Geometry is just geometry, isn’t it? Why is there a distinction between geometries? What makes Sacred Geometry so sacred?
In mathematics you can go in any direction with geometry, but every geometrical figure is created from a base. Actually, everything originates from a certain basis.
Sacred geometry is the basis of everything that is.
Everything? Why, then, do we not learn this?
Because this knowledge has been hidden for a very long time and most of it is forgotten. The complete knowledge seemed to be taught only to Priests in the so-called Mystery Schools, somewhere around the time of the ancient Egyptians (1500 B.C.). So it’s ” sacred” knowledge.
Unfortunately, we do not remember everything yet about sacred geometry and what is possible with it, but we are rediscovering more and more. Because geometry not only influences this dimension, but also other (higher) dimensions.
There is a lot of information to be found about sacred geometry and there is also a lot to tell about it, a complete study if you ask me. I wrote this blog to give some insight into why sacred geometry is so important.
What is the basis of everything that is?
Creation is the beginning of everything, without creation nothing can ever happen. In order to be able to create, consciousness is needed. Everything originated from consciousness according to the Mystery Schools.
So there is a form of consciousness in the complete emptiness. Movement is not possible, because you can only observe movement when you can move relatively to or from something. The only thing that was possible was to expand, and that can be done in 6 directions. The X, Y & Z directions, also called forward, backward, left, right, up and down. By expanding in all 6 directions, space has been created. And in that space you can create further.
By connecting all the ends of the directions, you create a figure, the octahedron. Movement is now possible, because there is an object where you can move relatively from one end to the other. Or you can move the object itself.
By rotating the octahedron around one of its own axes, the points of the octahedron “draw” a circle.
From this point on it becomes really interesting.
In sacred geometry, all straight lines are male energy, and all curved lines are female energy. So first a masculine form was created after which a feminine form was created from the masculine form, quite similar to the story about Adam and Eve.
Brief summary of everything so far; forms are created by a form of consciousness in a large infinite void. The whole story sounds very separate from everything, but it will take on more shape. (Pun intended)
The circle or sphere (seen in 3D) is the basis of all geometry and matter. Matter? Yes; protons, neutrons and electrons are spheres that together form an atom. A very simple example, but we will go into this in more detail. From the circle, all geometry can be further developed.
Lots of circles
Now we are going to do some complex stuff with circles, lots of circles. Because all these circles in the right proportion, are going to create the most important geometric shapes.
We start simply, with two circles. When two circles of equal size overlap with the centers of both circles on each other’s outer edge, the Vesica Piscis is created.
An important figure that is often used by churches and Freemasons, but also by the builders of the pyramid of Gizah and the Sphinx. The ratio of the outside of the pyramid can be traced back to the Vesica Piscis, which is linked to the golden section (Golden ratio, φ).
We will come back to this later.
Pyramid of Giza and the Sphinx
When the process is repeated with the number of circles placed, the “Seed of Life” is created. The Seed of Life is a figure consisting of 7 equal circles. When the correct lines are removed within the circles, the “Egg of Life” is created. The reason it is called the Egg of Life is because each fertilised egg with the first 8 cells of each living organism looks exactly like this.
Seed of Life
Egg of Life
If we continue to place circles, the Seed of Life becomes the “Flower of Life”. One of the most important figures in sacred geometry.
The Flower of Life is found all over the world in different temples. Among others in the temple of Seti I, the temple where Osiris has risen (resurrection) in Abydos. In different temples in Jerusalem and India and many more places in the world. All geometry can be traced back to the Flower of Life.
Flower of Life
Flower of Life, Jerusalem
From the Flower of Life, by removing the right lines again, the Fruit of Life can arise. When straight lines (male energy) are added to the Fruit of Life from the centers of the circles, Metatron’s Cube is created.
Metatron’s Cube contains the blueprints of all Platonic solids.
Fruit of Life
In short, the Flower of Life contains; the Vesica Piscis + Golden Ratio, the Seed of Life, the Egg of Life, the Fruit of Life, Metatron’s Cube, all Platonic Solids and the Tree of Life as well.
But what exactly is a platonic solid? A platonic solid is a three-dimensional figure that has to meet four requirements:
- All surfaces have the same shape and size
- All edges are equal in length
- The interior angles between each face is the same
- If the solid is placed inside a sphere, all the points are touching the surface of the sphere
The most well-known platonic solid is the cube: all planes are the same (square), all edges are the same length and all angles between the planes are the same (in the case of the cube 90 degrees).
In total there are 5 Platonic solids; the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. Sometimes the merkaba and the sphere itself are also considered to be platonic solids. All these figures can be found in Metatron’s Cube, as 3D figures with a 2D view.
The video below makes it a bit more clear how to find these figures.
You can read here about all the Secrets the Platonic Solids hold.
The Platonic solids were first described by the philosopher Plato (hence the name) around 400 BC.
But these remarkable shapes are being dug up all over the world and are thousands of years older than Plato.
The special thing about the platonic solids is that they represent every possible molecular connection and contain all possible geometrical laws of the reality and the universe in which we live, it is therefore the foundation of all physical matter.
Now you might wonder, where can I see these figures in matter? Sometimes it can be seen with the naked eye, sometimes it requires a microscope.
A good example is crystals and their growth patterns. Fluorite or pyrite, for example, a gemstone that grows perfect cubes. Fluorite, for example, has an atomic structure of an octahedron. Which means that when the fluorite breaks, it breaks into perfect octahedrons. Or ice crystals that always grow perfect hexagons, the 2D view of a cube.
There are also life forms that have a perfect platonic solid as their body, such as Radiolaria. Small single-celled organisms, some of which have a perfect icosahedron as their body.
Or a fertilized egg cell with 4 cells, together form a tetrahedron and with 8 cells a cube. But also DNA, the blueprint of all life, is made up of several dodecahedrons that together form the spiral.
Golden Ratio and Fibonacci
We had already mentioned it a moment ago, the Golden Ratio.
The golden section, Golden Ratio, Golden Mean, Divine Proportion, Phi, φ, are all names for the same, a ratio. And what about Fibonacci? That resembles it, but it is not the same thing.
The Golden Ratio is the ratio A (the largest part) / B (the smallest part) = (A + B) / A = φ
Nice and clear.
Phi(φ) is an infinite number, without a repetitive pattern (so completely random), comparable to Pi(π). Rounded off is φ; 1.61803398875… or also called 5^0.5 x 0.5 + 0.5.
So what can we do with that number? You can see it as a division. Suppose we have a line of 2 cm that we are going to divide with the Golden Ratio. The largest piece (A) becomes 2/φ = 1,236… What remains of this is piece B, so 2 – 1,236 = 0,763…
With this ratio we can make a Golden Ratio spiral. This spiral can continue indefinitely, both smaller and larger. Theoretically, this spiral has no beginning and no end.
But what can we do with this ratio?
The human brain is enormously attracted to this ratio and that is not surprising, because this ratio can be found everywhere. Our entire body, including organs, is made up of these proportions. And so too are bodies of animals, the entire nature of Earth and even our Universe, solar system, planets and Milky Ways. Take a look at our Pinterest page, there are a lot of nice examples.
Sacred Geometry in our structures
Because we are so attracted to this ratio, these ratios have also been incorporated into our art, as in buildings (Notre Dame, Greek Temples, the Pyramid of Giza), paintings, sculptures, etc. But nowadays, the Golden Ratio is also used in company logos and other graphic designs.
But especially the pyramid of Giza contains several special proportions.
The pyramid builders knew exactly what they were doing, because not only the pyramid itself is a masterpiece, but also the alignment with the other pyramids is in relation to the stars (Orion’s belt, among others), the number of the speed of light and the Golden Ratio.
The pyramid of Giza has a Φ½, 1, Φ triangle ratio, the same ratio as the Earth and the Moon have to each other. A bizarre coincidence! Which, of course, is no coincidence.
Squaring the Circle
The perimeter of the square has the same length as the circumference of the largest circle. This is called ‘Squaring the Circle’.
The size of the Moon and its midpoint, matches exactly with the outer circle of the Earth. Creating the ratio of the Pyramids of Giza.
Another master artist who also knew exactly what he was doing in his art is Leonardo Da Vinci (1452 – 1519). He incorporated so many proportions into his paintings that, after many years, every time something new is discovered in his art.
To make a representation of how many proportions there are in his painting, below is an analysis of one of his paintings.
But then the Fibonacci. Leonardo Fibonacci lived about 250 years earlier than Leonardo Da Vinci. Fibonacci discovered some interesting numerology. He saw that some lily species and irises had 3 petals. Butterflowers, delphiniums and akeletons had 5 petals, and other species had 8 again. He discovered that there was a pattern in it; he called it the Fibonacci sequence.
The series runs as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233…. Etc. The series has a beginning, but no end. Each subsequent number is added together with the two previous numbers. The Golden Ratio is also infinite but has no beginning and no end. And having no beginning is something that nature (and our thinking pattern) has a lot of trouble with.
The Fibonacci sequence is the solution of nature on the Golden Ratio (φ).
Golden Ratio and Fibonacci compared
But to what extent does the Fibonacci resemble the Golden Ratio? To be able to compare these two properly, we have to calculate the ratio of the Fibonacci series, because the Golden Ratio is also a ratio. This is done as follows:
|Current number||Previous Number||Division||Ratio||Phi (φ)|
|1||1||1 / 1||1.0||1.618034|
|2||1||2 / 1||2.0||1.618034|
|3||2||3 / 2||1.5||1.618034|
|5||3||5 / 3||1.666666||1.618034|
|8||5||8 / 5||1.600000||1.618034|
|13||8||13 / 8||1.625000||1.618034|
|21||13||21 / 13||1.615384||1.618034|
|34||21||34 / 21||1.619048||1.618034|
|55||34||55 / 34||1.617647||1.618034|
|89||55||89 / 55||1.618182||1.618034|
|144||89||144 / 89||1.617978||1.618034|
|233||144||233 / 144||1.618056||1.618034|
Now that the (different) proportions are visible, they can be compared with the Golden Ratio. To take you back, φ = 1,61803398875… As you may see, the ratio fluctuates enormously at the beginning of the sequence, from far below to far above the number φ. But these fluctuations become smaller and smaller as the sequence continues.
If you had a graph made of this, the Fibonacci series will come increasingly closer to Phi, but will never be exactly the number. Phi is also seen as the divine number/ratio because almost everything in our reality tries to approach this ratio. A justified name for “sacred” geometry.
Geometry and other dimensions
It is probably obvious now that everything is connected when it comes to sacred geometry and has a great influence on us and our dimension.
But sacred geometry goes beyond our dimension. It can create energy fields, attract vortexes and forms of consciousness. Maybe you have experienced this before, maybe you have not.
I work a lot with sacred geometry and it strikes me that everyone feels attracted to another form of sacred geometry and in everyone’s case geometry does something different for everyone.
Experiment with it, play and research it and above all, feel it. This is only a small grasp of sacred geometry.


March 2, 2023
The past few years have seen a flurry of interest in applications of the discrete Fourier transform (DFT) to both pitch and rhythmic domains of music analysis. Despite the fruitfulness of these approaches, the dense mathematics of the DFT can often be a deterrent to many theorists and analysts. In this talk, I focus on how the DFT can be used as an analytical lens to analyze the macroharmony of both notated scores and recorded performance. I show that the DFT is particularly adept at describing how different performances of the same work project harmony into the recorded track. Finally, I provide a tutorial on the DFT with the practical goal of making this tool accessible, approachable, and useful to a wider cohort of scholars.


This article describes the Production Mix Problem by way of an example and shows how to solve it using the Excel solver. Here we look at how many products to make of each kind when the products use some of the same components of limited availability.
Many manufacturers produce multiple products that are somewhat related and share some of the same components. In this case the Production Mix Problem occurs. This is a problem that can be formulated as linear program (LP) and solved to optimality. In this post we will give a minimal example in terms of the product and component number as well as the complexity of the problem.
The objective in the Production Mix Problem is to maximize the total profit over all products. The decision to be made is how much to produce of each product to achieve this. The constraint is to not exceed the amount of inventory for each of the needed components.
The input data needed to solve this problem are the quantities needed of each component for each product, the inventory of each component, and the profit per unit of each product. An example for the input data can be found in the table below.
In this example, the production of 1 unit of Product 1 requires 2 units of Component 1, 1 unit of Component 2 and 1 unit of Component 3 (none of Component 4). Selling one unit of product 1 will earn a profit of $20. There are 200 units of Component 1 in stock, 150 of Component 2 and so on.
|Components||Product 1||Product 2||Product 3||Product 4||Inventory|
|Profit per product||20||40||25||30|
Connecting the input data with the results through formulas
In the next step, a few formulas have to be entered into the spread sheet, see the screenshot below. This provides the Excel solver with important information about the problem. In the column UNITS USED, we compute the inventory for each component that remains after production of all planned units. Beforehand, when there is no entry for the decision variable yet, this will simply be zero, but will be updated as the solution is obtained. The INVENTORY AFTER is the difference between the inventory before and the units used. The PROFIT PER PRODUCT equals the product of profit per product and the quantity made for each product. In the cell for the total profit, these are added up over all products.
After these formulas are entered, the optimization problem can be solved using the following objective function and constraints. The objective function is the total profit which is the sum over all products of the quantity products per product multiplied by the profit for that product.
The constraints are that (1) the decision variable which is the number of products produced per product is integer and non-negative and (2) that the inventory needed to produce those quantities does not exceed the given inventory.
Entering the information into the Excel solver
The solver dialogue looks like shown in the graph below. The “Set Objective” (here $H$22) is the cell computing the total profit. The “Variable Cells” are the cells with the decision variables. In our case this is the number of products to be produced. Furthermore, there are constraints. The decision variable has to be both integer and non-negative which is expressed in the first two lines of the “Constraints”. The last line expresses that the used inventory must not exceed the inventory at hand.
Pressing solve in the above menu will give the following results using the Excel solver:
|UNITS USED||INVENTORY AFTER|
|Product 1||Product 2||Product 3||Product 4|
|NO. OF PRODUCTS MADE||48||22||34||34||TOTAL PROFIT:|
|PROFIT PER PRODUCT||$960||$880||$850||$1,020||$3,710|
The solution states that 48 units of Product 1, 22 units of Product 2, 34 units of Product 3, and 34 units of Product 4 should be produced to maximize the total profit which amounts to $3,710 in this example. The units used and the inventory after can be found in the column above as well.
Extensions and Remarks
The presented problem can be extended by several other constraints, for example by a minimum or maximum for the number of products made, or different profits depending on the quantity produced. Also alternate methods to produce a product (needing different amounts of components) is a possible extension of this problem.
A slightly different problem that has the exact same architecture arises when considering different machines instead of components. In that case there are several products that share machines. Each product takes a certain amount of time on each machine (can also be zero). The machine has limited capacity (before: inventory of the components) which is measured in time. Again, the objective is to find the profit maximizing production quantities for each product.
The mathematically identical problem, however with a somewhat different context occurs in the Production Mix Problem with Multiple Machines. The difference is that instead of products using the same components, these products share the same machines. And instead of inventory of components measured in pieces, we now have production capacity measured in time units (for example hours). You can read about it here. In that post, we used the exact same numerical example to highlight the similarities and differences.
If you would like this or a similar problem to be solved by us, please contact us.


Topologies for porous organic cages
In our article in Nanoscale 2017 (DOI: 10.1039/C7NR00703E) "Topological landscapes of porous organic cages", we define a nomenclature for the 20 most probable topologies for porous organic cage molecules, 12 of which have been reported so far.
Considering porous organic cages synthesised through pairs of building blocks via Dynamic Covalent Chemistry (DCC), we divide these into 4 families, dependent upon the number of reactive end groups: ditopic + tritopic, tetratopic + ditopic, tritopic + tritopic and tetratopic + ditopic. In the article, we discuss how some of these can be related to polyhedra, where examples of these topologies have been reported and how computer simulations can assist in predicting topological outcomes.
Tritopic + Ditopic family
Tritopic vertices are in blue, ditopic linkers in purple.
Tetratopic + Ditopic family
Tetratopic vertices are in yellow, ditopic linkers in purple.
Tritopic + Tritopic family
One of the tritopic precursors is in blue, the other in teal.
Tetratopic + Tritopic family
Tet6Tri8 topology. Tetratopic precursors are in yellow, tritopic in blue.
Our nomenclature for these topologies is described in the article as follows:
where X and Y are the two different component building blocks (BB) that constitute the cage. X and Y are labelled Di if they are ditopic, Tri if they are tritopic and Tet if they are tetratopic. The first BB, X, has the highest number of reactive end groups and if the underlying topology relates to a polyhedra, it will lie at its vertices. The second BB, Y, can have a number of reactive end groups less than or equal to X. Where the number of reactive end groups for X and Y is equal, which is denoted X and which is denoted Y is arbitrary. The superscripts, m and n, denote the number of each BB incorporated in the topology for X and Y respectively. The majority of the time, X-type BBs are connected to other X-type BBs through only one Y-type BB; in this case no subscript p will be given. However, if two X-type BBs are directly connected through links with two distinct Y-type BBs, then they have p = 2. The subscript, p, thus gives the number of double connections between BB pairs within a topology. These multiple links are also reported for periodic nets and result in topologies with multiple ring sizes. For some of the smaller topologies with only two X-type BBs, there is triple or quadruple linking of those BBs; in this case no subscript is given as there is no alternative connectivity for that topology. We finally suggest using the prefix c- in the case of catenated cages, equivalent to its usage as a prefix with interpenetrated periodic nets.
Topology vs geometric shape
Whilst we outline how the possible porous organic cage topologies can often be related to underlying polyhedra, such as Platonic solids, we note how this can cause confusion, leading to the same topology being named as different polyhedra, depending upon the geometric shape adopted by a specific molecule. In the Nanoscale article, we describe how a molecule with a single topology could adopt multiple different geometrical shapes, dependent upon factors such as the geometry of the component BBs and their conformational flexibility. For example, as shown in the figure below, a molecule with an underlying tetrahedral topology could have the approximate geometrical shape of a tetrahedron or an octahedron, or related intermediate shapes, dependent upon the component BBs:
The above figure is an example of how two porous organic cages that both have an underlying topology of a tetrahedron adopt different geometric shapes, in one case maintaining a tetrahedron shape (right hand side), and in the other adopting an octahedral shape (left hand side). The underlying topologies are shown in purple and the geometric shapes in orange.
In most cases, porous organic molecules of a single topology could adopt a whole range of shapes. For example, in the below figure, the range of geometrical shapes potentially accessible with a Tri4Di6 topology, from tetrahedron (left) to octahedron (right). The evolution of the shape is expressed through successive truncations of the vertices of the tetrahedron. The octahedron has been rotated into the page relative to the other shapes to give a clearer perspective. Bottom row: stages of truncation reproduced in hypothetical porous organic cages. The initial cage with a tetrahedral shape is shown in grey, precursors that lead to increasing truncation highlighted in teal, and the final (completely truncated) cage with an octahedral shape is in teal.


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Ingmar Visser - 2019-01-14 07:17 - depmixS4 - hidden Markov model classes
The 1.4 version of depmixS4 has an important (and much requested) new feature: the possibility to request standard errors of estimated parameters through the use of a finite differences approximation of the hessian. Comments and feedback much appreciated! After some time in this beta version we will post the new version to CRAN as well.
Peter Ruckdeschel - 2018-08-20 07:26 - RobASt - Robust Asymptotic Statistics
Updates for the packages of the RobASt family are now avaialable on CRAN in
Peter Ruckdeschel - 2018-08-20 07:25 - distr - S4 classes for distributions
Updates for the packages of the distr family are now avaialable on CRAN in
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Dear all, during the upcoming scheduled major maintenance work all of the services will be unavailable. The expected downtime is 5 JUN - 7 JUN. We regret any inconvenience that might cause. Best regards.
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Hi, everyone! I have some count and circle worksheets for you today.
I’ve been working on teaching my little one number recognition and kind of reinforcing their counting skills. These preschool counting worksheets will help your little one(s) learn and practice counting and recognizing numbers 1-3.
I’ve created more than thirty-five of these worksheets for you in different themes: summer, food and drink, sports, travel, winter, and more! You can download them all at once or you can download them by theme:)
Ways To Use These Worksheets
If your preschooler gets bored of circling the numbers, you can try mixing it up with one of these options:
- use cheez-it’s, M&M’s, or any other small edible pieces to cover up the correct numbers
- use finger paint to dot the correct number
- use do-a-dot markers to dot the correct number
- place the worksheet on a baking sheet or other magnetic surface and use small round magnets to cover the correct number
- change your utensils; use markers, color pencils, crayons, or highlighters
- laminate the worksheets and use a dry erase marker to circle or put an “x” on the correct number


NAME: Yang Wenyu
College of Science, Huazhong Agricultural University, Wuhan 430070, China
Shizishan Lu 1#, Hongshan, 430070, Wuhan, China
Ph.D. Computational Mathematics, Dalian University of Technology, China, 2008/9-2013/6
B.S. Information and Computing Science, Dalian University of Technology, China, 2004/9-2008/7
Machine learning, Neural Network, Bioinformatics.
2012 National Graduate Scholarship
2013 Outstanding Graduate of Dalian University of Technology
2013 Outstanding Graduate of Dalian City
<![if !supportLists]>1. <![endif]>Wenyu Yang, Jie Yang, Wei Wu, A Modified Spiking Neuron that Involves Derivative of the State Function at Firing Time, Neural Processing Letters Volume 36, Number 2 (2012), 135-144.
<![if !supportLists]>2. <![endif]>Wenyu Yang, Jie Yang, Wei Wu, A Modified One-Layer Spiking Neural Network Involves Derivative of the State Function at Firing Time, Lecture Notes in Computer Science, 2012, Volume 7367/2012, 149-158.
<![if !supportLists]>3. <![endif]>Jie Yang, Wenyu Yang, Wei Wu, A remark on the error-backpropagation learning algorithm for spiking neural networks, Applied Mathematics Letters Volume 25, Issue 8, August 2012, Pages 1118–1120.


The largest prime number and the largest perfect number were discovered
Prime numbers are unique in mathematics because they divide only by 1 and by itself. In other words, they can’t be broken down into smaller factors without remainders. Late last year, the international GIMPS initiative succeeded in finding the largest known prime number to date. It consists of more than 23 million digits, and its abbreviated name is M77232917. By the way, the largest perfect number known to date was also discovered.
The mathematical notation of the largest new prime number is 2,77232917 – 1 (two raised to the power of 77,232,917 minus 1). In total, it consists of 23 millionów 249 thousand 425 digits. Assuming that a standard A4 page holds almost 4 thousand charactersów, then to print the entire M77232917 it takes approx. 6 thousand pieces of paper! The new record holder has almost a million more digits than its predecessor.
Prime numbers fascinated the mathematicianóin as early as antiquity. Why are they so special? Because they divide only by 1 and by itself. Divided by any other number, they give a fractional result. Prime numbers include m.In.: 2, 3, 5, 7, 11, 13 or 17. During the centuryóIn many mathematicsóIn searched for larger and larger prime numbers.
One of the most prominent was the late 16th and early 17th century French monk Marine Mersenne. He studied the great prime numbers and discovered that someóre of them are in the form of powers of 2 minus 1. It is in his honor that today great prime numbers are classified as Mersenne numbers’a. M77232917 is the fiftieth number in this classification. Finding more causes more and more problems each timeów. It is predicted that 51. Mersenne number’and it will have more than 100 millionóin the digits.
The discovery of the largest known prime number today was made on December 26, 2017 by electronics engineer Jonathan Pace of Germantown, Tennessee (USA). The man has been involved in the search for large prime numbers for 14 years as part of the GIMPS program. The name of the project is skrót from Great Internet Mersenne Prime Search (with j. ang.: The Great Internet Search for Mersenne Prime Numbers’a).
The GIMPS project was launched in 1996. From https://www.mersenne.org/ anyone wishing to do so can download specialized software (big prime hunter) for searching for Mersenne prime numbers’a. Not an interferenceóIt ca work of the computer, because it only uses the computing power not used at the moment. With this solution, sixteen Mersenne prime numbers have been found in more than 20 years’a. What’s more, GIMPS is offering a reward of 3,000 to the lucky finder. dollarów. For finding the number Mersenne’a, whichórej the number of digits will be greater than 100 millionóa prize of 150 thousand is predicted in. USD.
The GIMPS project is overseen by Prof. Chris Caldwell, whoóry collects róalso stories of discoveriesów. It took an Intel I5-6600 CPU computer six days of non-stop operation to verify that M77232917 is indeed a prime number. However, in order to be certain of the discovery, the calculation needed to be confirmed on four róThey are used in computers and in four róof different programs. Each time, their work took from 37 to 82 hours.
In other words, each computer had to multiply by itself 77,222,917 dwóyek, subtract the number 1 from the resulting number and check that the resulting number actually divides only by 1 and by itself, giving no remainder.
Also associated with prime numbers are perfect numbers, whichóre are równe sum of its divisorsów. The smallest known perfect numbers are 6 and 28. The number 6 is divisible by 1, 2 and 3 (divisors). 1+2+3=6. According to the ancients, perfect numbers had magical properties. In the 18th century, mathematician Leonhard Euler proved the relationship between prime numbers Mersenne’a and the perfect numbers. Just take any prime Mersenne number’a (the nth power of 2 minus 1) and multiply it by 2 to the power of "n-1", a we get a perfect number. This means that with the discovery of the largest known prime number to date, we have also learned the largest known perfect number.


September Equinox (First day of fall.) in North America is on Thursday, September 22, 2016 at 10:21 AM EDT.
While we're on the subject of facts and figures ......, when you shuffle a deck of cards chances are that no one has ever shuffled them in that order before!
e think card games are pretty limited because there are only 52 cards, but it’s ridiculous how many combinations you have in these 52 cards. There are, of course, 52! possible combinations (remember the factorial thing above?), which is a humongous number:
That number is beyond astronomically large, but that’s exactly how many ways you can arrange 52 cards. So when you’re shuffling a deck, shuffle it properly – and chances are you will create a completely new arrangement, one that no one ever created before.


People are often very surprised by results like this:
>>> 1.2-1.0 0.199999999999999996
and think it is a bug in Python. It’s not. It’s a problem caused by the internal representation of floating point numbers, which uses a fixed number of binary digits to represent a decimal number. Some decimal numbers can’t be represented exactly in binary, resulting in small roundoff errors.
In decimal math, there are many numbers that can’t be represented with a fixed number of decimal digits, e.g. 1/3 = 0.3333333333…….
In base 2, 1/2 = 0.1, 1/4 = 0.01, 1/8 = 0.001, etc. .2 equals 2/10 equals 1/5, resulting in the binary fractional number 0.001100110011001…
Floating point numbers only have 32 or 64 bits of precision, so the digits are cut off at some point, and the resulting number is 0.199999999999999996 in decimal, not 0.2.
A floating point’s repr function prints as many digits are necessary to make eval(repr(f)) == f true for any float f. The str function prints fewer digits and this often results in the more sensible number that was probably intended:
>>> 0.2 0.20000000000000001 >>> print 0.2 0.2
Again, this has nothing to do with Python, but with the way the underlying C platform handles floating point numbers, and ultimately with the inaccuracy, you’ll always have when writing down numbers as a string of a fixed number of digits.
One of the consequences of this is that it is dangerous to compare the result of some computation to a float with == ! Tiny inaccuracies may mean that == fails. Instead, you have to check that the difference between the two numbers is less than a certain threshold:
epsilon = 0.0000000000001 # Tiny allowed error expected_result = 0.4 if expected_result-epsilon <= computation() <= expected_result+epsilon: ...


I have continued my pursuit of learning to play the piano. This second year of lessons has been going well. I focused more on developing good technique and timing. I’m feeling more and more comfortable when playing the piano.
Along with improving my skills, I’m starting to learn more about music theory. It is really interesting to learn about music theory. I never realized how much math is involved in music.
I keep meaning to post videos of myself playing. Unfortunately, I keep putting it off. 🙁


An implicit surface in three-dimensional space is defined as a level surface of a function f(x,y,z). That is, the surface consists of the set of points (x,y,z) such that f(x,y,z)=k, where k is a constant. As a simple example, the unit sphere can be defined as an implicit surface using the function f(x,y,z) = x2+y2+z2 and the constant k = 1.
An implicit surface can be unbounded, but 3D-XplorMath-J will show only the part of a surface that lies within a specified sphere centered at the origin. The radius of this sphere is given by the exhibit parameter Radius of Search Sphere. Recall that the value of this parameter, like all exhibit parameters, can be set using the "Set Parameters" command in the "Settings" menu.
3D-XplorMath-J provides two different visual representations of an implicit surface: as a dot-cloud and as a ray-traced image. The default representation is a dot-cloud rendered in the Anaglyph Stereo Vision. The user can select between these two representation using the "Use Point Cloud Rendering" and "Use Ray-traced Rendering" options in the "Actions" menu.
In the dot-cloud rendering, the implicit surface represented by a random scattering of points on the surface. The points are generated by selecting lines at random and calculating the points where those lines intersect the surface. (It is an interesting fact that -- provided the correct probability distribution is used on the set of lines -- the points chosen in this way are scattered evenly on the surface; that is, the number of points in a given piece of the surface is proportional to the area of that piece.) The number of desired surface points is specified by the parameter named "Number of Points in Point Cloud." The maximum number of random lines that will be checked is specified by the "Number of Random Lines" parameter. The program generates lines and computes intersection points until it has either found the desired number of points or checked the specified maximum number of lines.
In the ray-traced rendering, the representation of the surface is created by "casting rays": For each pixel in the image, consider the line from the viewpoint through that pixel. Determine the first point, if any, where this line intersects the surface. That point is the point that is visible at the given pixel, and the pixel should be colored to represent that point on the surface. The actual color that is used is computed using the normal vector to the surface at that point and the properties of the lights that are illuminating the surface. (The lighting properties can be set using the "Light Settings" command in the "Settings" menu). Ray-tracing produces a very realistic looking image.
Note that it can take some time to compute the full ray-traced image. When a ray-traced rendering first appears, a rough, low-resolution image is shown. The full-resolution rendering is then filled in progressively, as it is computed. The low-resolution image is also used when a ray-traced surface is being rotated or dragged.
(The Implicit Surfaces gallery is a sub-gallery of the Surfaces gallery, but it is quite a bit different from the other sub-galleries of that galleries.)


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The earliest recorded systems of weights and measures originate in the 3rd or 4th millennium Before the Common Era.
Can you tell how many lentilles wide the sluice is? A picture proof would be good here, just to know how you counted it.
Don't forget to use the tools given by the Pink Chest Keepers.
Send the number to the Pink Chest Keepers, write it down as a backup in your book and turn to the next page.
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Many women, as well as men, struggle with their weight and are unsure what an ideal weight is relatively to their height. By using this calculator, one can view an approximate estimate of their BMI showing whether they are underweight, average or overweight. Note, this calculator is an estimate as it only considers the height of one. For an exact measure, one must visit the doctor/specialist for a more precise calculation (online calculations don’t distinguish the ratio between fat and muscle).
One can calculate this:
1. Online at http://www.bmi-calculator.net
2. By using the formula:
|ENGLISH BMI FORMULA|
|BMI = ( Weight in Pounds / ( Height in inches x Height in inches ) ) x 703|
|METRIC BMI FORMULA|
|BMI = ( Weight in Kilograms / ( Height in Meters x Height in Meters ) )|


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Lights! Camera! Math!
February 24 @ 7:00 pm - 8:00 pm| $16, $14, $9
Danny long thought that school, and in particular math, wasn’t for him. All he wanted to be was a famous magician! But when he knuckled-down to learn the tricks of the trade, he realized that math was not only essential, it was the SECRET to a whole load of magic! Lights! Camera! Math! is a spectacular show where math, magic, theatre and digital technology collide to create a fun, interactive performance which proves understanding math can be fun!
Written by Paul Bosco Mc Eneaney
Produced by Cahoots NI, Belfast, Northern Ireland


7 edition of Exploring Mathematics with your Computer (New Mathematical Library) found in the catalog.
August 14, 1997 by The Mathematical Association of America .
Written in English
|The Physical Object|
|Number of Pages||311|
How to predict and calculate the positions of stars, planets, the sun, the moon, and satellites using a personal computer and high school mathematics. Our knowledge of the universe is expanding rapidly, as space probes launched decades ago begin to send information back to earth. There has never been a better time to learn about how planets, stars, and satellites move through the heavens.
The kiss of Judas
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Comparative reflections on the past and present political, commercial, and civil state of Great Britain
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Preliminary northeast U.S. market assessment, appendix F
Numerical derivatives and nonlinear analysis
Mystery of Totem Pole Inlet
Myths of Oz
Exploring mathematics with your computer (New mathematical library) by Arthur Engel (Author) ISBN ISBN X. Why is ISBN important. ISBN. This bar-code number lets you verify that you're getting exactly the right version or edition of a book. The digit and digit formats both : $ Description This is a mathematics Exploring Mathematics with your Computer book, not a programming book, although it explains Pascal to beginners.
It is aimed at high school students and undergraduates with a strong interest in mathematics, and teachers looking for fresh ideas. It is full of diverse mathematical ideas.
About the book: This is a mathematics book, not a programming book, although it explains Pascal to beginners. It is aimed at high school students and undergraduates with a strong interest in mathematics, and teachers looking for fresh ideas.
It is full of diverse mathematical ideas requiring little background. The main point of this book is to introduce a new tool for exploring math, and to foster its use by applying it to a variety of topics. In other words, it provides a ramble through a variety of topics rather than a systematic study of any one of them.
This is a mathematics book, not a programming book, although it explains Pascal to beginners. It is aimed at high school students and undergraduates with a strong interest Exploring Mathematics with your Computer book mathematics, and teachers looking for fresh ideas.
It is full of diverse mathematical ideas requiring little background. “Mathematics for Computer Science” from MIT OpenCourseWare is available in pdf format. Download lecture notes, assignments, exams and solutions.
Book Description. This course offers an introduction to Discrete Mathematics oriented toward Computer Science and Engineering. Buy Exploring Mathematics with your Computer by Arthur Engel from Waterstones today.
Click and Collect from your local Waterstones or get FREE UK delivery on orders over £ Buy Exploring Mathematics with your Computer by Arthur Engel from Waterstones today. Mathematics The goal of computational mathematics, put simply, is to ﬁnd or develop algo-rithms that solve mathematical problems computationally (ie.
using comput-ers). In particular, we desire that any algorithm we develop fulﬁlls four primary properties: • Accuracy. An accurate algorithm is able to return a. Exploring Mathematics gives students experience with doing mathematics - interrogating mathematical claims, exploring definitions, forming conjectures, attempting proofs, and presenting results - and engages them with examples, exercises, and projects that pique their interest.
including mathematics and statistics, computer science. The aim of this book is to present some the basic mathematics that is needed by computer scientists. The reader is not expected to be a mathematician and we hope will find what follows useful.
Just a word of warning. Unless you are one of the irritating minority mathematics is hard. You cannot just read a mathematics book like a novel. My Books; Browse Exploring Mathematics with Your Computer > Top Shelves Top shelves for Exploring Mathematics with Your Computer Showing of 6 to-read.
7 people. my-library. 1 person. digital. 1 person. mathematics. 1 person. computer-science. 1 person. Your next choice is 50 Visions of Mathematics () edited by Sam Parc, which was published for the fiftieth anniversary of the foundation of The Institute of Mathematics and its Applications (IMA).
This book, appropriately, has 50 articles in it. It was edited by. Brother you are to send me pdf links of an algebra mathematics. love.I request to you. love.I wait. Reply Amar February 8, at am. About the book Exploring Math for Programmers and Data Scientists showcases chapters from three Manning books, chosen by author and master-of-math Paul Orland.
You’ll start with a look at the nearest neighbor search problem, common with multidimensional data, and walk through a real-world solution for tackling it. Turtle Geometry presents an innovative program of mathematical discovery that demonstrates how the effective use of personal computers can profoundly change the nature of a student's contact with mathematics.
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Apple. Android. Windows Phone. Android. To get the free app, enter your mobile phone : Bolster. A typical computer science program contain following mathematics related courses 1.
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So you might be wondering why writing is required in your math class now. The Greek word mathemas, from which we derive the word mathematics, embodies the. Go Math. Standards Practice Book Grade 5 Go Math.
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This is so because the core content of Mathematics is the same around the world. Math Adventures with Python will show you how to harness the power of programming to keep math relevant and fun. With the aid of the Python programming language, you'll learn how to visualize solutions to a range of math problems as you use code to explore key mathematical concepts like algebra, trigonometry, matrices, and cellular automata.
Get this from a library. Exploring math from algebra to calculus with Derive: a mathematical assistant for your personal computer. [Jerry Glynn]. An Illustrated Guide to Exploring Math with Code. by Peter Farrell.
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From Building Blocks (National Science Foundation), ideas for finding the mathematics in and developing mathematics from children's activities. National Council of Teachers of Mathematics (NCTM) offers math standards, Principles and Standards for School Mathematics, and many activities, web-based software environments, and videos.
Axiom is a free general purpose computer algebra system. The book gives a technical introduction to AXIOM, interacts with the system's tutorial, accesses algorithms developed by the symbolic computation community, and presents advanced techniques. ( views) Mathematics for Computer Scientists by Gareth J.
Janacek, Mark L. Close - BookBoon, Here is an unordered list of online mathematics books, textbooks, monographs, lecture notes, and other mathematics related documents freely available on the web. I tried to select only the works in book formats, "real" books that are mainly in PDF format, so many well-known html-based mathematics web pages and online tutorials are left out.
W hen I was a college student, I saw a list of essential math books on a blog. I promised to myself to read all those books in 10 years because there were 50 books on that list. I am still trying. Exploring Mathematics: Investigations for Elementary School Teachers is a text designed to give readers a highly conceptual understanding of mathematics topics essential for Common Core Standards-based elementary school teaching.
In this book, the development of mathematical concepts, methods, and skills takes place through discussions and analyses of alternative solutions to engaging Reviews: 2.
The book seems to have been written in a similar vain to C. Stanley Ogilvy's "Excursions through Geometry". i.e. it shows up some of the beauty of the subject in a reasonably clear and concise way, without being too high s: 6. Tip: It's easier to find your test banks if you save them to the TestGen/tests folder on your computer.
That way, when you choose Open Book from the TestGen File menu, the test banks files are immediately visible for you to select. Opening a TestGen test bank. Start the TestGen application.
From the File menu, select Open Book. You can browse. Apologia Educational Ministries offers a full range of homeschool science curriculum, from elementary courses through advanced high school materials, founded upon a biblical, creationist viewpoint.
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This is probably one of the best books on mathematics books I have ever read. Everything is perfect. The writing, the typography, the exercises, the cover design. And it is written by Knuth.
Exercises are slightly harder than usual but you'll always find easier exercises. Don't let it break your motivation. The official 4-H store for pins, record book covers, club supplies, educational curriculum, and apparel for youth and adults. Proceeds support the 4-H mission.
ECS is pleased to announce the release of. An Electronic Textiles Unit for Exploring Computer Science. In our new curriculum unit, students explore electronic textiles (e-textiles): articles of clothing, accessories, or home furnishings with embedded electronic and computational curriculum is an alternate for Unit 6: Robotics.
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Numbers play a big part in the HVAC industry. We don’t just mean the difference between a comfortable indoor 68 degrees and a too-cool 45 or muggy 95. No, an HVAC technician has to be familiar with a lot of other numbers and calculations to help make those comfortable conditions possible.
There’s a lot of tools you can use to get the right numbers, like checking reference books or using an app. But it’s also important to understand how they work.
Knowing some of the basic formulas involved in HVAC work will help you work faster and more reliably. While there’s nothing wrong with relying on other sources, you should be able to make some calculations yourself – both during your certification exams and in the field.
Here’s some of the most important formulas you should know!
E= voltage (emf)
I = amperage (current)
R = resistance (load)
Ohm’s law states:
E = I x R which also means:
I = E/R
P = E x I
or if your P is in kilowatts:
P = (E x I)/1000
U = 1/R That is, if R = 4, U = ¼, or 0.25
3 Phase Voltage Unbalance
= (100 x maximum deg. from average volts) / Average Volts
One Farad capacity: 1 amp stored under 1 volt of pressure
MFD (microfarad) = 1 MFD = 1 Farad / 1,000,000
= 6.24 X 1018 (1 Coulomb = 1 Amp)
VA (how the secondary transformer is rated) = volts x amps
Work and Horsepower
Work = force x distance
So if you weigh 180 lb, and climb stairs 100 ft high, the work is 180 lb x 100 ft = 18,000 ft-lbs (foot pounds)
1 Horsepower = 33,000 ft-lb of work in 1 minute
HP to KW to BTU
1 HP = 746 Watts
1 KW =3413 BTU
Ton of Refrigeration
The amount of heat to melt 1 ton of ice at 32 degrees F. This amount is 12,000 BTU per hour, which is 288,000 BTU in 24 hours
RPM of motor = (60Hz x 120) / (No. of Poles)
1800 RPM Motor – slippage makes it about 1750
3600 RPM Motor – slippage makes it about 3450
What’s in the Air?
Dry Air = 78.0% Nitrogen 21.0% Oxygen 1.0% Other Gases
Wet Air = Same as dry air plus water vapor
Specific Density = 1 / Specific Volume
Specific Density Of Air = 1 / 13.33 = .075 lbs./cu.ft.
Standard Air = 24 Specific Heat (BTU’s needed to raise 1 lb. 1 degree)
Heat and Humidity
Enthalphy = h = Sensible heat + Latent heat
Relative Humidity = Moisture present / Moisture air can hold
Specific Humidity = Grains of moisture per dry air 7000 GRAINS in 1 lb. of water
Dew Point = When wet bulb equals dry bulb
Total Pressure (Ductwork) = Static Pressure + Velocity Pressure
28 Inches Of Wc (water column) = 1 psi
Furnace Efficiency: % Efficiency = energy output / energy input
Useful HVAC Formulas
For Conditions Other Than Standard Air
Total Heat (BTU/hr) = 4.5 x cfm x Δh (std. air)
Sensible Heat (BTU/hr) = 1.1 x cfm x Δt (std. air)
Latent Heat (BTU/hr) = 0.69 x cfm x Δgr. (std. air)
Total Heat (BTU/hr) = 500 x gpm x Δt (water)
GPM cooler = (24 x TONS) / Δt (water)
Fluid Mixture Tm = (Xt1 + Yt2) / X + Y (this works for air or water)
BTU/hr = 3.413 x watts = HP x 2546 = Kg Cal x 3.97
Lb. = 453.6 grams = 7000 grains
psi = ft. water/2.31 = in. hg/2.03 = in. water/27.7 = 0.145 x kPa
Ton = 12,000 BTU/hr = 0.2843 x KW
HP (air) = cfm x Δp (in.H2O)/6350 x Eff.
HP (water) = gpm x Δp (ft.)/3960 x Eff.
gpm = 15.85 x L/S
cfm = 2.119 x L/S
CFM = ______BTU/Hr / ( 1.08 x Temperature Difference)
Therm = 100,000 BTU = MJ/105.5
Watt/sq. ft. = 0.0926 x W/M²
PPM (by mass) = mg/kg


At their December 5 meeting, Falmouth School Committee members wrestled with what has been a recurring issue in the district.
Why are students' English MCAS scores meeting or exceeding the state average while math scores are not?
"I just find it mind boggling that we don't see the same achievement in math," school committee chairman Kelly Welch said during a presentation on spring MCAS results for children in grades 3 through 8. "I'm just very worried about it."
Zachary Waddicor, director of curriculum and instruction for the district's four elementary schools, said that the spring results represent students' first time taking the "Next Generation MCAS", which places a greater focus on problem solving and critical thinking and better gauges students' "college or career readiness." He said he expects that scores will improve in the coming years.
Students' progress in English Language Arts [ELA] and math were tracked in two categories: achievement and growth. Achievement tracks student scores, while growth measures students' progress in both subject areas compared to students' statewide who shared the same scores in past tests.
Results show that between 54 and 68 percent of students met or exceed expectations in English in grades 3 through 8. Students tested better than the state average at each grade level.
Students in grades 4, 5, and 6 showed noticeable growth greater than the state average, while 47 percent of students in grade 7 showed growth on the test from last year, just shy of the state average of 50 percent. Grade 8 students narrowly topped the state average in growth at 50.5 percent.
Waddicor also noted that students with disabilities, high needs, and economic disadvantages all achieved ELA scores greater than the state average at each grade level.
"What's really powerful is students in all of our subgroups are performing beyond what the state is," he told the committee.
In math, student achievement hedged closer to the state average, while third grade students performed below the state average. Scores show that 41 percent of third grade students scored proficient or better in math, this compared to the state average of 49 percent.
In terms of growth, students in grades 4, 6, and 7 bested the state average. In fifth grade, 48 percent of students showed growth in Math compared to the state average of 50 percent. The gap was wider in eighth grade, where 39 percent of students showed growth compared to the state average of 50 percent.
"Obviously we have the ELA part figured out, but math is not there," Welch said.
Committee member Terri Medeiros asked how growth statistics ultimately impact the district. Waddicor said that the figures do not have any impact on whether or not a student graduates. Scores in achievement, however, do reflect on the district and students.
Going forward, Waddicor and Sonia Tellier, the district's assistant superintendent of curriculum and instruction, said administrators plan to coordinate more professional development to help students make gains in MCAS math. Administrators could also implement more engaging methods to help bring students up to speed, including introducing a problem of the month for students across the district to work on.
"We want to put a proactive, positive spin on it," he said.
Welch encouraged efforts to more creatively teach math to students in the district.
"The kids need to see the passion for math that they see for language," she said. "Building that passion will translate over to the kids, I'm sure."
The committee also reviews science, technology, and engineering scores for students in grades 5, 8, and 9. The percentage of fifth and ninth grade students who scored proficient or advanced was above the state average in each grade. In eighth grade, 37 percent of students scored in the proficient or advanced categories, compared to the state average of 40 percent.


NEW MADISON--Tri-Village High School has received a significant contribution to their Science, Technology, Engineering, and Mathematics (STEM) program. The STEM program is pushing the boundaries of technology in the classroom, opening up opportunities for students to experience new computer, engineering, and design programs they would not otherwise encounter. Thanks to Matt Fraylick and IBM, Tri-Village was honored to receive a grant for $3,000 to support the program and purchase resources for the students.
Utilizing funds from the IBM grant, Tri-Village Jr./Sr. high students have been actively engaged in mathematics, science and art! The school was able to purchase a cube 3D printer. Students have been learning 3D printing software to design and print 3D models in ABS plastic. Students are designing plastic parts for local businesses as prototypes, crafting jewelry, and creating polyhedrons then printing them on the printer. Students have been able to implement a variety of their mathematics skills, learn vital technology skills, and learn problem solving throughout the process. Most of the students are motivated simply by the ability to print their virtual creation in plastic. It has been a great learning tool that has enhanced their curriculum and significantly improved student motivation towards learning mathematics, science, and technology.
Tri-Village was also able to fund a Wacom tablet. This new technology tool has significantly increased their ability to attract creative students into their STEM program. The tablet is used as a drawing/creating surface for artwork, textures for game design levels and objects, and 3D modeling. Students wait in line to get their hands on this very impressive tool, just to do their “engineering” work.
The $3,000 grant from IBM has allowed the TV STEM program the opportunity to increase student motivation and significantly enhance our learning of mathematics, science and technology. Tri-Village is deeply honored to be receiving the same grant for next year’s program, and greatly appreciate IBM's support in their efforts to improve STEM at Tri-Village.
|The school was able to purchase a cube 3D printer. Students have been learning 3D printing software to design and print 3D models in ABS plastic.|


A Mathematicians Lament.
"Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soulcrushing ideas that constitute contemporary mathematics education. ... The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such. Everyone understands that poets, painters, and musicians create works of art, and are expressing themselves in word, image, and sound. In fact, our society is rather generous when it comes to creative expression; architects, chefs, and even television directors are considered to be working artists. So why not mathematicians?" (via).


Title: SOME ALGEBRAIC PROPERTIES OVER THE GENERALIZED GENERAL PRODUCT OBTAINED BY MONOIDS AND GROUPS
Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814
Article ID: MTJPAM-D-19-00003; Volume 1 / Issue 1 / Year 2019, Pages 96-106
Document Type: Research Paper
Author(s): Suha A. Wazzan a
aDepartment of Mathematics, KAU King Abdulaziz University, Science Faculty, Girls Campus, 21589, Jeddah-Saudi Arabia
Received: 9 July 2019, Accepted: 4 September 2019, Available online: 28 November 2019.
Corresponding Author: Suha A. Wazzan (Email address: email@example.com)
Full Text: PDF
Suppose A and B be arbitrary monoids. In , it has been recently defined a new consequence of the general product A⊕B δ⋈ψ B⊕A under the name of the generalized general product, and then has been given a presentation for it. In this paper we give some algebraic properties of the generalized general product obtained by some certain monoids and groups.
Keywords: General product, Left cancellative monoids, Bands, Equidivisible, OrthodoxReferences:
- G. Baumslag, Wreath products and finitely presented groups, Math. Z. 75, 22-28, 1961.
- A.S. Cevik, S.A. Wazzan, F. Ates, The generalized of general products of monoids. (In review).
- N.D. Gilbert, S. Wazzan, Zappa-Szép products of bands and groups, Semigroup Forum 77, 438-207, 2008.
- J.B. Hickey, M.V. Lawson, Unit regular monoids, Proc. Edinburgh Math. Soc. 127A, 127-144, 1997.
- J.M. Howie, Fundamentals of semigroups theory, Clarendon Press, Oxford, 1995.
- J.M. Howie, N. Ruskuk, Construction and presentations for monoids, Comm. in Alg. 22 (15), 6209-6224, 1994.
- M. Kunzo, Zappa Products, Acta Math. Hung. 41, 225-239, 1983.
- T.G. Lavers, Presentations of general products of monoids, Journal of Algebra 204, 733-741, 1998.
- M.V. Lawson, Inverse semigroups : The theory of partial symmetries, World Scientific, 1998.
- M.V. Lawson, A correspondence between a class of monoids and self-similar group actions I, Semigroup Forum 76, 489-517, 2008.
- M.V. Lawson, A correspondence between a class of monoids and self-similar group actions II, Inter.J. Algeb. Comp. 25 (04), 633-668, 2015.
- R.B. McFadden, Unit-regular orthodox semigroups, Glasgow Math. J. 25, 229-40, 1984.
- J.D.P. Meldrum, Wreath products of groups and semigroups, Longman, Harlow, 1995.
- V. Nekrashevych, Self-similar groups, 117, American Mathematical Soc., 2005.
- S. Oguz, E.G. Karpuz, Some semigroup classes and congruences on Bruck-Reilly and generalized Bruck-Reilly ⋆-extensions of monoids, Asian-European J. Math. 8 (4), 1550075 (11pages), 2015.
- N.U. Ozalan, A.S. Cevik, E.G. Karpuz, A new semigroup obtained via known ones, Asian-European J. Math. 13 (1), 2040008 (13 pages), 2020.
- J. Szép, On factorisable, not simple groups, Acta Univ. Szeged. Sect. Sci. Math. 13, 239-241, 1950.
- G. Zappa, Sulla construzione dei gruppi prodotto di due sottogruppi permutabili tra loro, Atti Secondo Congresso Un. Mat. Ital. Bologna, 119-125, 1940.


Owen & Eli: 32 Months
This week we worked on Nn is for Numbers. Lots of potential, kind of a bust on execution. The boys were pretty good with number recognition but not very good or interested in actually counting. By the end of the week, Eli was counting from 1-10 outloud (but saying 10 instead of 6 every time) and Owen, when asked to count, would say 'no, not count, Eli do it.' I'm not going to push them, they'll get it eventually!
They used Cheerios to cover up the number that I would call out, and then used the same page to match number magnets. They put bean bags on their Sesame Street flashcards and used their number magnets on the number review sheet from Confessions of a Homeschooler. We practiced counting using 1:1 ratio with pompoms on the same sheet, glass beads in an ice cube tray with number stickers, and buttons on pipe cleaners. Hit or miss, but the repetition was helpful, I'm sure.
We're on to Oo next week, but I'm planning on keeping numbers a part of our weekly activities from now on. To see what other Tot Schoolers did this week, check out 1+1+1=1
Tomorrow is the last day to enter my felt food pattern giveaway! You can't win if you don't enter : )


The Golden Mean, the Fibonacci sequence, or Fractals are all mathematical terms for these uncanny patterns in nature that are so universal they seem to be a fundamental characteristic of the Universe. The Fibonacci sequence starts like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 and so on forever. Each number is the sum of the two numbers that precede it.
These patterns appear throughout nature in the arrangements of flower petals, seed heads, pine cones, tree branches, shells, spiral galaxies, and faces. They can seem too perfect to be true, but they occur in nature and plants all the time. Even at the microscopic level we see the Fibonacci sequence. The DNA molecule measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. These numbers, 34 and 21, are numbers in the Fibonacci series. Look at the number series in the paragraph above and you will see it.
These patterns and relationships have delighted artists, scientists, philosophers and mathematicians for thousands of years. Galileo Galilei wrote, “[The universe] is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures.” Photographers have captured these forms in breath taking images and here are over 20 of them.


The Star Academy’s primary objective is to capture a child’s natural curiosity and to develop his/her potential as a lifelong learner.
Our private school offers Preschool, Elementary School, and Middle School.
Star Academy has two locations in Watertown, MA and Wellesley, MA.
Our customized curriculum combines proven traditional American methods with the world’s best innovative approaches to teaching. We offer Preschool, Pre-K, Kindergarten, Elementary School (Grades 1-5), Middle School (Grades 6-8) and a convenient after-school program.
Our early learning program is aimed at developing flexibility of children’s minds and utilizing their intellectual potential to the fullest.Learn More
Learn more about our academically-forward elementary school program for Grades 1-5. Find course offerings, schedule, curriculum, and more.Learn More
Learn more about our dynamic, academically rigorous, middle school program for Grades 6-8. Find course offerings, schedule, curriculum, and more.Learn More
Learn more about our specially formulated Summer Camp options at our Watertown location! Find course offerings, schedule, curriculum, and more.Learn More
The Founder and Director of Star Academy, Larisa Bankovsky, is a proven math teacher of over 33 years. She has experience teaching students of all school levels in both former Soviet Union and in the United States. Larisa holds two Master's Degrees - Master's in Mathematics and Teaching Mathematics from Dnepropetrovsk State University (Ukraine) and Master of Arts in Education from City College of City University of New York. She has been developing mathematical curriculum as well as specialized programs for advanced mathematical development of students for over twenty years.


Shahzeb is a SAMBa PhD student in mathematical biology with a background in applied mathematics. His work focuses on cell migration and the collective behaviour of cells.
Shahzeb earned his BSc in Mathematics from the University of Greenwich. He then went on to do an MSc, also in Mathematics, at the Queen Mary University of London, where he worked with Dr Rosemary Harris on stochastic models with memory effect. He joined SAMBa in September 2019 and subsequently joined the Yates Group in May 2020.
In common with the rest of the group, his work concerns modelling biological phenomena using stochastic agent-based models. He is particularly interested in modelling the migration and collective movement of the melanoblast neural crest cells in the early stages of embryo formation. He is also working on adding realism to existing cell models by allowing more complex dynamics (such as interactions between multiple species of cells) and deriving the corresponding deterministic models.
Aside from academia, when not in the pandemic, Shahzeb is an avid badminton player, likes to read fiction, cook and learn to play the violin. He also loves cats.


Teach One, Learn One is the teacher professional development component of Making Awesome Things Happen (M.A.T.H.) 501c3 nonprofit organization.
M.A.T.H. is a nonprofit organization dedicated to developing mathematical proficiency among traditionally underrepresented students (African American, Hispanic, Native American, and Economically Disadvantaged). M.A.T.H. also aims to increase the number of women and minorities interested in and pursuing STEM careers.
M.A.T.H. provides professional development to train, develop, and equip teachers, especially teachers in schools with high populations of traditionally underrepresented students, to help students be successful in mathematics and academics overall. We provide research based, hands-on, nontraditional training opportunities related to all things mathematics, topics that effect students learning of mathematics, and culturally relevant and social justice pedagogies. M.A.T.H. strongly believes in implementing culturally relevant and social justice pedagogies to create independent and woke (alert to injustice in society) learners.
Feel free to contact us at email@example.com if interested in our on-site professional development opportunities.
Meet The Dream Team
Dr. Katina Thomas
Doctor of Education in Curriculum and Instruction with a concentration in English Language Arts and Reading, Master of Education in Administration and Supervision, and Bachelor of Science in Interdisciplinary Studies
Dr. Johnitha Johnson
Ph.D. in Curriculum and Instruction with an emphasis in Urban Education, Master of Science in Juvenile Forensic Psychology, Bachelor of Science in Psychology
Master of Science in Applied Mathematics, Bachelor of Science in Mathematics
Dr. Sharon Deloach
Doctor of Education in Educational Leadership and Policy, Master of Education in Education Administration and Curriculum and Instruction, Bachelor of Science in Health Science Administration and Education Generalist
Dr. LaToya Austin
Ph.D. in Curriculum and Instruction with an emphasis in Mathematics Education, Master of Engineering in Industrial Engineering, and Bachelor of Science in Applied Mathematical Sciences
Master of Education in Educational Administration and Supervision, Bachelor of Science in Health Services
Master of Education with an emphasis in mathematics, Bachelor of Science in Electrical Engineering
Dr. Tarcia Hubert
Ph.D. in Curriculum and Instruction with an emphasis in Mathematics Education, Master of Science in Mathematics, Bachelor of Science in Mathematics


How likely am I to win the new national lottery?
This week, Camelot, the operator of the National Lottery announced an increase in the amount of balls in the lottery by 10, an increase to 59 balls. As an occasional lottery player, I wondered how much worse this made the odds. How much harder would it be to win the jackpot?
As it turns out, the odds of winning the 59 ball jackpot decrease from 1 in 13,983,816, under a 49 ball lottery, to 1 in 45,057,474 under the 59 ball format. This makes it over 3 times as difficult to secure the life-changing jackpot.
The team at Figure It Out had a go at calculating and collecting some of the odds of lotteries around the world to see if the UK is getting a raw deal.
World’s hardest jackpots to win
But turns out the UK is following a global trend; some of the world’s largest lotteries were also found to have recently decreased their odds of winning the jackpot.
Lotteries around the world found guilty of making it harder to win the jackpot:
So, how are these odds calculated? Let’s say you have the jackpot winning lottery ticket in a 49 ball and 6 number lottery. Your first winning number is picked among 49 balls, each of them having an equal chance of being picked. Each lottery ticket has 6 numbers, the odds of the first ball matching one of the numbers on your ticket are therefore 49 divided by 6. The next ball is now drawn from the 48 remaining and there are only 5 unmatched numbers on your ticket. This goes on up until we find out the sixth winning number where at this point there are 44 remaining balls and only one number left on the ticket.
By multiplying these numbers together we get our jackpot odds of:
Now by adding an additional 10 balls to the mix the new chance of winning the jackpot in the new 59 ball national lottery is:
Does this mean I will stop buying the occasional lottery ticket?
In fact, Camelot claims that the new lottery format gives players a better chance of becoming a millionaire. How could this be so? There is an additional change to the lottery which in fact does make this the case. There will be an additional raffle that will award prizes of £1m and £20,000. The odds of winning this will vary weekly as it depends on the number of entries, however Camelot claims that odds to become a millionaire have therefore increased from 1 in 14 million to approximately 1 in 10 million by winning either the jackpot or the £1 million pound prize. The new lottery format means bigger rolling jackpots, as individuals are less likely to win it, but more chances of becoming a millionaire due to the raffle.
The greedy amongst us will feel hard done by, with the prospect of winning £20 million looking far less likely. However, could we settle for the increased odds of us winning a measly £1 million? Personally, I’ll probably still buy the occasional ticket - but let’s face it, it’s still more likely that we’ll be struck by lightning than become a lotto millionaire...
Specific details on the World’s hardest jackpots to win
Source: Odds are calculated from guess range from official website of the respective lotteries
The specific details of recent changes to lotteries:
This post is a follow-up to “It could be you, but it probably won’t be by Shivam Desai and Jon Chadwick”
This list has been mentioned in national newspaper Daily Express.


AC's attempts at learning to do this in R.
Added package lme4 to R. http://cran.r-project.org/web/packages/lme4/lme4.pdf
Also added plyr package, because it has a rename() function for renaming columns in a data frame: (http://www.cookbook-r.com/Manipulating_data/Renaming_columns_in_a_data_frame/)
However, I found it easier to do this:
based on guidance from:
Column names can be given explicitly via the col.names; explicit names override the header line (if present).
Added arm package: http://cran.r-project.org/web/packages/arm/index.html
arm: Data Analysis Using Regression and Multilevel/Hierarchical Models
R functions for processing lm, glm, svy.glm, merMod and polr outputs.
arm package provides the
coefplot function for plotting the regression coefficient values and std. errors.
Need to specify that subjects is a nominal variable, using the
Tell R that a variable is nominal by making it a factor. The factor stores the nominal values as a vector of integers in the range [ 1… k ] (where k is the number of unique values in the nominal variable), and an internal vector of character strings (the original values) mapped to these integers.
A nice description from:
In this case the coefficient corresponds to one of the
terms in the model and I would advocate performing a likelihood ratio
test comparing the two models
>fm <- glmer(SameSite~BreedSuc1+Sex+(1|Bird), family="binomial") >fm0 <- glmer(SameSite~Sex+(1|Bird), family="binomial") # the null >hypothesis model >anova(fm0, fm) >
Even though the function is called anova it will, in this case,
perform a likelihood ratio test (LRT). It also prints the values of
AIC and BIC if you prefer to compare models according to one of those
criteria but I prefer using the likelihood ratio for nested models.
However, before doing that comparison you should ask yourself whether
you want to compare models that have the, apparently unnecessary term
for Sex in them. The way I would approach the model building is first
to reduce the model to
>fm1 <- lmer(SameSite~BreedSuc1+(1|Bird), family="binomial") >
You could then compare
>anova(fm1, fm) >
which I presume will give a large p-value for the LRT, so we prefer
the simpler model, fm1. After that, I would compare
>fm2 <- lmer(SameSite ~ 1 + (1|Bird), family="binomial") >


Seminars & Colloquia Calendar
Generic Regularity for All Minimal Hypersurfaces in 8-Manifolds
Zhihan Wang (Princeton University)
Date & time: Tuesday, 25 October 2022 at 2:50PM - 3:50PM
Abstract: The well-known Simons cone suggests that singularities may exist in a stable minimal hypersurface in Riemannian manifolds of
dimension greater than 7, locally modeled on stable minimal hypercones. It was conjectured that generically they can be perturbed away. In this talk, we present a way to eliminate these singularities by perturbing metric in an 8-manifold. By combining with a Sard-Type Theorem for space of singular minimal hypersurfaces of dimension 7, joint with Yangyang Li, we proved that in an 8-manifold with generic metric, every locally stable minimal hypersurface has no singularity.
In particular, this proves the existence of infinitely many SMOOTH minimal hypersurfaces in a generic 8-manifold.


It's been a while since I wrote on my dyscalculia and how it affects me not only when I'm knitting or crocheting, but also in my every day life with all the challenges it brings.
From my earliest memories of school, numbers have always scared me. In Primary 3, when I was about seven years old, we were given times tables on sheets of paper that we had to cut up, put in an envelope, shake about and then remove and reassemble in the correct order. I remember struggling with the 6,7,8 and 9 times tables for the entire year to the point where I was incapable, crying in class and at home over how I couldn't do the calculations required to come up with the correct answer. I eventually managed to give the right answers, but it was based on memorising it; "4x6=24" because I remember that's the answer, not because I can work it out. For the rest of my time at Primary school, my maths work would always come back with "Show your working!" written in the margin. Concurrently, I excelled at reading and writing, so beyond me not being very good at mathematics, there was nothing to indicate I had any learning difficulties.
Fast-forward to Secondary school and my mathematics did not advance. To explain the Scottish school system at the time, there are six years of Secondary school, and during your 2nd year, you choose what subjects you want to take over the next two years, but are also assessed so you can learn the subjects at an appropriate difficulty. This also dictates what level your final exams will be - if you were in Credit, it meant you took "Level 1 or 2". General was 3/4 and Foundation was 5/6. If you were any lower than 6, you were given assistance with learning. I was taking nine subjects, eight of which were Credit with one being Foundation. What was that Foundation? Maths, of course.
As Secondary school progressed, the difficulties I have with numbers became more apparent in other classes. I had taken biology and graphic communication, which during the first couple of years of Secondary school had been general in their content - there were some numbers in there, but there was enough other content for me to guddle through the classes. However, I was firmly within Foundation maths, and the moment I could drop the subject altogether, I did (in favour of Philosophy, which went on to be my best exam result). Foundation maths was really just a repeat of the first couple of years of Secondary school, and even covered a lot of content from Primary school. There was no long division, no complex fractions and almost no algebra at all. It was so long ago that I took my exams that the past papers from that year are not available online, but I did have a look at more recent Foundation Mathematics past papers, and the level of questions are pretty much what I remember them being:
Work out the answers to the following.
(a) 6427 + 125
Eve is paid £7·50 per hour.
(a) How much is Eve paid for working 4 hours?
The next one is my favourite:
George is going to knit a sweater.
He needs to buy 10 balls of wool and 2 pairs of knitting needles.
One ball of wool costs £3.
One pair of knitting needles costs £2·50.
How much will it cost George to knit the sweater?
When I told people that a question in my exam paper was "Count the petals on this flower - are they odd or even?", no one believed that a 15 year old could struggle with such a basic concept. That I was doing so well in other classes indicated that I just wasn't trying hard enough when it came to maths, with the old trope of "If you're really good at English and bad at maths, it's because you hate maths and would rather be doing something else" cropping up again and again.
This is one of the most frustrating aspects of having dyscalculia - the constant "You just need to practice" followed up with "Come on, it's not that hard" that I'm met with whenever I forgo a coping technique I've developed and try to work something out in a "legitimate" way. I struggle with reading travel time tables and calendars (which extends into understanding how much time has passed between two points on the clock), volume, depth, length, distance, sequences, and even dialing phone numbers. Often people claim that much of what they learned at school hasn't translated into their everyday life - calculating the length of something isn't needed, so it's not much of a loss to no longer be able to do it. However, there is a big difference in my experiences with numbers and someone who's a bit rusty - I couldn't do it in the first place, and no amount of practice will ever help me learn it either. Basic mathematical skills are used daily in ways that most people take for granted, whether it's filling the car up with petrol or knowing when to leave the house to make sure they get to work in time. These are things that I often mess up, and have to put my own working method in place in order to successfully carry out simple tasks.
So how does all of this relate to crochet and knitting? Well, the most obvious factor here is that both crafts involve a lot of numbers. The big difference here, though, is that unlike a problem on paper, I can hold fabric in my hand, and counting stitches is like an extension of counting my fingers (something that I still do). I wasn't able to be taught how to do either growing up because my mother would assume I was able to keep count of things as I was going, and found that any attempt to teach me focused too much on the numbers and pattern, rather than grasping the basics like how to hold the needles and yarn, how to wrap yarn or even how to centre-pull a ball of yarn so that it doesn't keep rolling away. When I took up the fiddle a few years ago, I was taught that learning how to hold a violin and bow was more important in the early stages than reading music. The same is true for knitting and crochet, and is one of the core principles of my workshops for beginners - don't worry if you can't read patterns for now, that can come later after you've gained confidence in the tools and materials.
Learning how to read patterns was a process that also took many years, and was a driving force in writing my own patterns; I would have to deconstruct many patterns and re-write them using my own terminology, then translate them again into industry standard terminology. I have to not only double-check everything I do, but triple-check, make more than one swatch and often make numerous prototypes of an item. Decoding charts was a huge leap forward too; being more visual, I find it easier to follow pictures than numbers. I try to encourage new crocheters to engage with charts as early as they feel capable, because it by-passes much of the mental arithmetic present in a written pattern. I can place my finger on a chart and trace it, then do the same with the fabric in my hand. While the numbers and totals in written patterns may appear to be self-explanatory, to me they can be daunting and misread. Part of my number blindness, that I touched on further up, is my tendency to incorrectly dial phone numbers. I can look at a printed number and instantly forget it, meaning that it's very difficult for me to retain what a total should be in my head while also counting up or down. With charts, I find it less stressful to keep count. As much as I prefer charts, I do feel a great sense of accomplishment when I write a pattern that can be followed by others. I find that if I don't knit or crochet for a couple of days, my overall ability and confidence with numbers drops significantly, and affects other areas of my life, even if it's something as simple as reading the numbers on a measuring jug.
So there you have it - a brief foray into what it's like to be a knitter and crocheter when you have trouble with numbers. I hope I've been able to share some experiences that others can relate to, and maybe even get folks talking about the less well known challenges the world of numbers has to those of us who aren't wired to deal with it.


Syed Ejaz Alam ( PMRC Research Centre, Jinnah Postgraduate Medical Centre, Karachi. )
The Variability of observations (Standard deviation)mean only defmes the general position of the distribution and other values of the distribution that are scattered or spread around it cannot be predicted by it. For this reason the mean alone is of limited value. The range is an important measurement as the figures at the top and bottom denote the trends as removed from generality. Range fails to give much indication of the spread of observations about the mean. This is where the standard deviation comes in. The theoreticalbasis of the standard deviation is complex but whenever a distribution is normal the standard deviation provides a useful basis for interpreting data. Many biological characteristics conform to normal distribution closely enough for it to be commonly used1.
Figure shows the normal (Gaussian) distribution calculated from diastolic blood pressures of 500 men mean 82 mm Hg standard deviation 10mm Hg. The standard deviation is useful because if observations follow a normal distribution then mean ±ISD includes about 68% and mean ± 2SD covers about 95% of observations. However, the standard deviation can be calculated from the observations and the steps are as follows2:
1. List the observations and calculate the mean.
2. Alongside the listed observations write the difference between each value in turn and the mean.
3. These figures are now squared and added up, producing the sum of squares.
4. This sum of squares is divided by one less than the number of observations to produce the mean sum of squares or variance.
5. By taking the square root of the variance one obtains the standard deviation, which can lie on either side of the mean. Calculations using the height of ten subjects for standard deviation are shown in Table.
The standard deviation is usually shown as mean ± 2S.D.
This mean and the scatter of values around it as described by two standard deviations is a much better way for expressing the results than just giving the range of values.
The two values which describe the limits of two standard deviations in either direction from the mean are 95% probability limits. Normal values for the laboratory data are commonly expressed in this way. In clinical pharmacology blood levels of drugs from limited numbers of subjects should also show the standard deviation to give the best idea of the total
scatter of results. The number of estimations should also be given as for example, for 20 estimations of the serum protein: serum protein 6.7 ± 1.7 W100 ml2.
Standard error (S.E.):
It is the inherent variation from one sample to another. It shows the variability that the value would show if repeated samples are drawn from the same population. It is calculated by the formula: S.E = S.D /J~
where S.D. is the standard deviation of the observations and n is number of observation.
1. Siddiqui, M.A. Role of Statistics in Medical Research. Pakistan Medical Research Council, pp. 30-37.
2. Hawkins, C., Sorgi, M. Research How to Plan, Speak and Write About it. R.J. Acford. Terminus Road Industrial Estate Chichester Sussex 1985; pp 140-141.


2018/19 Statistics Creative-Writing Competition for Secondary School Students
This competition is co-organised by the Hong Kong Statistical Society and the Education Bureau.
The key objectives of the competition are:
- to raise the interest of students in statistics and its application; and
- to encourage students to creatively express in words the daily application of statistical concepts or put statistical concepts into a story in a scientific and objective manner.
2018/19 Prize Presentation Ceremony on 22 June 2019
2018/19 Briefing Seminar on 27 October 2018 (Saturday)
- Introduction of rules and adjudication criteria (Chinese only)
- Introduction of “Statistics in Sports” (Chinese only)
- 2017/18 champion team sharing (Chinese only)


This week, we begin a series to help you become better acquainted with each department at Maranatha.
Helping our students navigate through equations, calculations, and critical thinking are the talented faculty of our Math Department. The department’s goal is to help students develop the problem solving and critical thinking skills necessary to be successful in current and future STEM classes and in life.
The team is led by Monica Chan, the department’s chair, who has been teaching at Maranatha for 8 years, and loves cooking, sleeping, traveling, and watching Korean dramas. The department includes many other amazing teachers. Cheryl Dixon ‘80 is a Maranatha alum, and a faculty member for 9 years, who enjoys traveling. Also teaching for 9 years in our Math Department is another alum, Kimi Nahigian ‘08, who is a fan of line dancing. Beloved veteran teacher, Sandy Pelletier, has prepared our students for careers in mathematics and related fields for 30 years, and in her free time enjoys swimming, kayaking, paddle boarding, and biking. Michael Garcia is in his fourth year at Maranatha and likes to garden and make/can fruit jam. Ji Choo joined the team 3 years ago and enjoys playing softball. The newest member of the team, Joel Staggers, is enjoying his 2nd year with the department and can sometimes be found on the soccer field when he’s not in the classroom.
With a self-proclaimed love for God, Math, and their students, these devoted teachers work hard to help their students improve their math skills and find interesting methods to keep students engaged. A few of the special projects they have planned for this year include: Celebrating Pi Day on March 14, 2022; Financial Algebra students will theoretically purchase a car with $45,000; Honors Math Analysis students will create their own math podcast; Geometry students have a project titled “Ames Room” (an optical illusion room); and Honors Multivariable Calculus students will do a mathematical exploration of microphones (including omnidirectional mics, cardioid mics, and hyper cardioid mics), and will write polar equations for each model to determine the percent of sound picked up in front of and behind the microphone, so they may compare the results and conclude how the microphones differ.
Maranatha is blessed to have these talented, creative teachers educating and challenging our students.