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Learn MoreDo you know that ** CLOCKS **add numbers in a different way than we do? Do you know that

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.

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.

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

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. ðŸ™‚

Passion for Mathematics.

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