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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

I was trying to come

up with something

new to work on, logical,

but not numerical. So

I started making these

drawings and switching

the arrows different

ways. (See pic below, click on)

Can someone give me

ideas as to what I might do

next with this and what

branch of mathematics

do you think this might be

close to.

Also imagine a cube with

8 boxes in 3-D and having

arrows go in six coordinate

directions.

Thanks for helping me

from boredom to the pursuit

of awesome things...

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

If the arrows are allowed to go East, West, South and North.

Then there are 4^4 or 256 permutations.

Then if you account for quarter rotations 4 times around,

then there are 70 shapes according to my BASIC program

I wrote this morning. Still I have to draw them out from

the program results to check for bugs.

Interesting that 70 is greater than 256 / 4.

I suspected that from other similar projects, such as

shapes in boolean algebra cubes of 3-D. (3 variables)

**igloo** **myrtilles** **fourmis**

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

There are more than 64 combinations because rotating doesn't always get you 4 different things.

For example, Up Right Left Down wouldn't be affected by rotation at all.

Now I'm curious as to how many distinct combinations there would be if you counted two things as the same when they're a reflection of each other.

Why did the vector cross the road?

It wanted to be normal.

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**soroban****Member**- Registered: 2007-03-09
- Posts: 452

. .

. . You go ahead . . . I'll wait in the car.

.

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

To mathsy,

Your question about the reflection reduction is

a good one. That is the same as looking through from the

backside of the piece of paper of the drawings.

My groupings on a large pad of paper show there are 45

of these arrangements. And 45 > (70 / 2).

(Hi soroban, I might tackle your questions next week.... Sweet ideas!)

**igloo** **myrtilles** **fourmis**

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**Avon****Member**- Registered: 2007-06-28
- Posts: 80

I would usually use Burnside's lemma to count such things.

In the first example we are letting the rotation group of the square act on the drawings. This group has four elements: the identity, the quarter-turn clockwise, the quarter-turn anticlockwise, and the half-turn.

The identity fixes all of the 256 drawings.

The quarter-turns each fix 4 drawings.

The half-turn fixes 16 drawings.

Hence the number of different drawings if two are considered the same if one is a rotation of the other is

(256+4+4+16)/4 = 70.

If we also want to allow reflections then there are four more elements in our group: the two reflections in the diagonals of the square, the reflection in the vertical line through the centre of the square, and the reflection in the horizontal line through the centre of the square.

The two diagonal reflections don't fix any of the drawings.

The reflections in the vertical and horizontal lines each fix 16 drawings.

Hence the number of different drawings if reflections and rotations are considered the same is

(256+4+4+16+0+0+16+16)/8 = 39.

I notice that this is not the same as what John counted.

The 2x2x2 cube could be handled similarly, but I don't really have the time to do it right now.

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

Fascinating Avon!! This is amazing

there are ways to calculate this count

from the 1800's. I don't get it yet

though, but thanks!!

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

Avon, I found

an easy non-mathematician's

article on burnside's lemma.

http://baxterweb.com/puzzles/burnside5.pdf

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

My 45 count was incorrect, done too hastily, and I missed 6 pairs.

Avon is correct about the 39 arrangements, which includes

reducing by rotations and reflexions (reflections).

**igloo** **myrtilles** **fourmis**

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**hero_hont****Member**- Registered: 2010-08-19
- Posts: 2

John E. Franklin wrote:

My 45 count was incorrect, done too hastily, and I missed 6 pairs.

Avon is correct about the 39 arrangements, which includes

reducing by rotations and reflexions (reflections).

hih, i count 44.. but i don't know how do this exercises do solution?

have you reply total this exeercises solution...i can reference it?

thanks......

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

Maybe if I get the time, I will

draw up or use LaTeX to

draw up the arrangements.

If I do that, I will try

to put pairs that are

reflections beside each

other, so you will see how

the 70 pair up to 39.

**igloo** **myrtilles** **fourmis**

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**hero_hont****Member**- Registered: 2010-08-19
- Posts: 2

hih.............:P

I will have started making these

drawings and switching

the arrows different

ways.

hih, try work............

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

Yes, I hope you

draw them out and

get 39 and 70, because I

am busy with reading

"The Trachtenberg

Speed System Of Basic Mathematics"

book right now...

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

I'll try a little LaTeX, hope I can get the 2 by 2 array up...

Okay, I got the LaTeX to work.

The eight boxes of arrows below are

the only eight ones from the 70 that

cannot be paired up with a mirror

image one. So there are 31 pairs (62),

plus 8 non-pairs. 31 + 8 = 39 arrangements.

I have not drawn out the 31 pairs yet.

Hope I don't have to...

(Keep in mind we are doing quarter rotations 3 times, and

consider those all the same with the 70 count)

*Last edited by John E. Franklin (2010-09-02 01:13:35)*

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

So it appears the numbers

70 and 39 are a trend in

this thread. Does anyone

know something in

chemistry that has these

two numbers and look

similar?

**igloo** **myrtilles** **fourmis**

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