Categories

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.

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?

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.

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.

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

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

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.

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. 🙂

Passion for Mathematics.

Categories

Natural Geometry of Natural Numbers

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.

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.

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.

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:

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:

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

The numbers are eagerly waiting for your comments. 🙂

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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