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College Mathematics I.S.I. and C.M.I. Entrance Math Olympiad USA Math Olympiad

Clocky Rotato Arithmetic

Do you know that CLOCKS add numbers in a different way than we do? Do you know that ROTATIONS can also behave as numbers and they have their own arithmetic? Well, this post is about how clock adds numbers and rotations behave like numbers. Let’s learn about clock rotation today

Consider the clock on earth.

clock

So, there are 12 numbers {1,2, …, 12 } are written on the clock. But let’s see how clocks add them.

What is 3+ 10 ?

Well, to the clock it is nothing else than 1. Why?

Say, it is 3 am and the clock shows 3 on the clock. Now you add 10 hours to 3 am. You get a 13th hour of the day. But to the clock, it is 1 pm.

So, 3 + 10 = 1.

If you take any other addition, say 9 + 21 = 6 to the clock ( 9 am + 21 hours = 6 pm ).

Now, you can write any other Clocky addition. But you will essentially see that the main idea is :

The clock counts 12 = 0.

Isn’t it easy? 0 comes as an integer just before 1, but on the clock, it is 12 written. So 12 must be equal to 0. Yes, it is that easy.

Cayley’s Table

This is a handsome and sober way to write the arithmetic of a set. It is useful if the set is finite like the numbers of the CLOCK Arithmetic.

Let me show you by an example.

Consider the planet Cheenta. A day on Cheenta consists of 6 earth hours.

So, how will the clock on Cheenta look like?

Cheenta Planet clock

Let’s us construct the Cayley Table for Cheenta’s Clocky Arithmetic. Check it really works as you wish. Here for Cheenta Clock, 3 = 0.

Clock rotation explanation

Exercise: Draw the Cayley Table for the Earth (24 hours a day) and Jupiter (10 hours a day).

Nice, let’s move on to the Rotato part. I mean the arithmetic of Rotation part.

Let’s go through the following image.

clock rotation

Well, let’s measure the symmetry of the figure. But how?

Well, which is more symmetric : The Triskelion or the Square (Imagine).

clock rotation

Well, Square seems more right? But what is the thing that is catching our eyes?

It is the set of all the symmetric positions, that capture the overall symmetry of a figure.

For the Triskelion, observe that there are three symmetric operations that are possible but that doesn’t alter the picture:

  • Rotation by 120 degrees. \(r_1\)
  • Rotation by 240 degrees. \(r_2\)
  • Rotation by 360 degrees. \(r_3\)

For the Square, the symmetries are:

  • Rotation by 90 degrees.
  • Rotation by 180 degrees.
  • Rotation by 270 degrees.
  • Rotation by 360 degrees.
  • Four Reflections along the Four axes

For, a square there are symmetries, hence the eyes feel that too.

So, what about the arithmetic of these? Let’s consider the Triskelion.

Just like 1 interact (+) 3 to give 4.

We say \(r_1\) interacts with \(r_2\) if \(r_1\) acts on the figure after \(r_2\) i.e ( 240 + 120 = 360 degrees rotation = \(r_3\) ).

Hence, this is the arithmetic of the rotations. To give a sober look to this arithmetic, we draw a Cayley Table for this arithmetic.

Treskelion

Well, check it out.

Exercise: Can you see any similarity of this table with that of anything before?

Challenge Problem: Can you draw the Cayley Table for the Square?

You may explore this link:- https://www.cheenta.com/tag/level-2/

And this video:- https://www.youtube.com/watch?v=UaGsKzR_KVw

Don’t stop investigating.

All the best.

Hope, you enjoyed. πŸ™‚

Passion for Mathematics.

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College Mathematics I.S.I. and C.M.I. Entrance Math Olympiad USA Math Olympiad

The Organic Math of Origami

Did you know that there exists a whole set of seven axioms of Origami Geometry just like that of the Euclidean Geometry?

Related image
The Origami in building the solar panel maximizing the input of Solar Power and minimizing the Volume of the satellite

Instead of being very mathematically strict, today we will go through a very elegant result that arises organically from Origami.

Before that, let us travel through some basic terminologies. Be patient for a few more minutes and wait for the gem to arrive.

In case you have forgotten what Origami is, the following pictures will remove the dust from your memories.

Origami Dinosaurs
Origami Papers

Origami (from the Japanese oru, β€œto fold,” and kami, β€œpaper”) is a traditional Japanese art of folding a sheet of paper, usually square, into a representation of an object such as a bird or flower.

Flat origami refers to configurations that can be pressed flat, say between the pages of a book, without adding any new folds or creases.

Non- Flat Origami
Image result for flat origami
Flat Origami

When an origami object is unfolded, the resulting diagram of folds or creases on the paper square is called a crease pattern.

We denote mountain folds by unbroken lines and valley folds by dashed
lines. A vertex of a crease pattern is a point where two or more folds intersect, and a flat vertex fold is a crease pattern with just one vertex.

The Origami and the Crease Pattern

In a crease pattern, we see two types of folds, called mountain folds and valley folds.


Related image

Now, if you get to play with your hands, you will get to discover a beautiful pattern.

Related image
The blue lines denotes the mountain folds and the red lines indicate the valley folds.

The positive difference between the dotted lines and the full lines is always 2 at a given vertex. The dotted line denotes the mountain fold and the full line denotes the valley fold.

This is encoded in the following theorem.

Maekawa’s Theorem: The difference between the number of mountain
folds and the number of valley folds in a flat vertex fold is two.

Isn’t it strange ?

Those who are familiar graph theory may think it is related to the Euler Number.

We will do the proof step by step but you will weave together the steps to understand it yourself. The proof is very easy.

Step 1:

Let n denote the number of folds that meet at the vertex, m of which
are mountain folds and v that are valley folds, so that n = m + v. (m for mountain folds and v for valley folds.)

Step 2:

Consider the cross section of a flat vertex.

Cross Section of a Flat Vertex

Step 3:

Consider the creases as shown and fold it accordingly depending on the type of fold – mountain or valley.

Folding the Cross Section of a Flat Vertex along the creases

Step 4:

Now observe that we get the following cross section. Observe that the number of sides of the formed polygon is n = m + v.


Step 5:

We will also count the angle sum of the n sided polygon in the following way. Consider the polygon formed.

We get a four sided polygon here with internal angles 0 degree and 360 degrees.
An upper view of the four sided triangle formed.

Observe that the vertex 2 is the Valley Fold and other vertices are Mountain Fold. Also the angle subtended the vertex due to valley fold is 360 degrees and that of due to the mountain fold is 0 degrees.

Therefore, the sum of the internal angles is v.360 degrees.

Step 6:

Now we also know that the sum of internal angles formed by n vertices is 180.(n-2), which is = v.360.

Hence we get by replacing n by m+v, that m – v = 2.

QED

So simple and yet so beautiful and magical right?

But it is just the beginning!

There are lot more to discover …

Do you observe any pattern or any different symmetry about the creases or even in other geometry while playing with just paper and folding them?

We will love to hear it from you in the comments.

Do you know Cheenta is bringing out their third issue of the magazine “Reason, Debate and Story” this summer?

Do you want to write an article for us?

Email us at babinmukherjee08@gmail.com.







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College Mathematics I.S.I. and C.M.I. Entrance Math Olympiad USA Math Olympiad

Natural Geometry of Natural Numbers

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How does this sound?

The numbers 18 and 30 together looks like a chair. 

The Natural Geometry of Natural Numbers is something that is never advertised, rarely talked about. Just feel how they feel!

Let’s revise some ideas and concepts to understand the natural numbers more deeply.

We know by Unique Prime Factorization Theorem that every natural number can be uniquely represented by the product of primes.

So, a natural number is entirely known by the primes and their powers dividing it.

Also if you think carefully the entire information of a natural number is also entirely contained in the set of all of its divisors as every natural number has a unique set of divisors apart from itself.

We will discover the geometry of a natural number by adding lines between these divisors to form some shape and we call that the natural geometry corresponding to the number.

Let’s start discovering by playing a game.

Take a natural number n and all its divisors including itself.

Consider two divisors a < b of n. Now draw a line segment between a and b based on the following rules:

  • a divides b.
  • There is no divisor of n, such that a < c < b and a divides c and c divides b.

Also write the number \(\frac{b}{a}\) over the line segment joining a and b.

Let’s draw for number 6.

Number 6 has the shape like a square

Now, whatever shape we get, we call it the natural geometry of that particular number. Here we call that 6 has a natural geometry of a square or a rectangle. I prefer to call it a square because we all love symmetry.

What about all the numbers? Isn’t interesting to know the geometry of all the natural numbers?

Let’s draw for some other number say 30.


The Natural Geometry of 30 – A Cube

Observe this carefully, 30 has a complicated structure if seen in two dimensions but its natural geometrical structure is actually like a cube right?

The red numbers denote the divisors and the black numbers denote the numbers to be written on the line segment.

Beautiful right!

Have you observed something interesting?

  • The numbers on the line segments are always primes. 

Actually, it shows that to build this shape the requirement of the line segments is as important as the prime numbers to build the number.

Exercise: Prove from the rules of the game that the numbers on the line segment always correspond to prime numbers.

Did you observe this?

  • In the pictures above, the parallel lines have the same prime number on it.

Exercise: Prove that the numbers corresponding to the parallel lines always have the same prime number on it.

Actually each prime number corresponds to a different direction. If you draw it perpendicularly we get the natural geometry of the number.

Let’s observe the geometry of other numbers.

Try to draw the geometry of the number 210. It will look like the following:

Image result for Is four dimensional hyper cube graph

The natural geometry of the number 210 – Tesseract
Image result for tesseract cube
This is the three dimensional projection of the structure of the Tesseract.

Obviously, this is not the natural geometry as shown. But neither we can visualize it. The number 210 lies in four dimensions. If you try to discover this structure, you will find that it has four different directions corresponding to four different primes dividing it. Also, you will see that it is actually a four-dimensional cube, which is called a tesseract. What you see above is a two dimensional projection of the tesseract, we call it a graph.

A person acquainted with graph theory can understand that the graph of a number is always k- regular where k is the number of primes dividing the number.

Now it’s time for you to discover more about the geometry of all the numbers.

I leave some exercises to help you along the way.

Exercise: Show that the natural geometry of \(p^k\) is a long straight line consisting of k small straight lines, where p is a prime number and k is a natural number.

Exercise: Show that all the numbers of the form \(p.q\) where p and q are two distinct prime numbers always have the natural geometry of a square.

Exercise: Show that all the numbers of the form \(p.q.r\) where p, q and r are three distinct prime numbers always have the natural geometry of a cube.

Research Exercise: Find the natural geometry of the numbers of the form \(p^2.q\) where p and q are two distinct prime numbers. Also, try to generalize and predict the geometry of \(p^k.q\) where k is any natural number.

Research Exercise: Find the natural geometry of \(p^a.q^b.r^c\) where p,
q, and r are three distinct prime numbers and a,b and c are natural numbers.

Let’s end with the discussion with the geometry of {18, 30}. First let us define what I mean by it.

We define the natural geometry of two natural numbers quite naturally as a natural extension from that of a single number.

Take two natural numbers a and b. Consider the divisors of both a and b and follow the rules of the game on the set of divisors of both a and b. The shape that we get is called the natural geometry of {a, b}.

You can try it yourself and find out that the natural geometry of {18, 30} looks like the following:

Looks like a chair indeed!

Sit on this chair, grab a cup of coffee and set off to discover.

The numbers are eagerly waiting for your comments. πŸ™‚

Please mention your observations and ideas and the proofs of the exercises in the comments section. Also think about what type of different shapes can we get from the numbers.

Also visit: Thousand Flowers Program

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