# Square Cubes

1. Apr 15, 2004

### Russell E. Rierson

Is it possible to compress a 3D object into 2 dimensions?

For example:

1 + 2 + 3

2 + 3 + 4

3 + 4 + 5
___________

6 + 9 + 12 = 3^3 = 27

Here is a "square" 6^3

1+2+3+4+5+ 6
2+3+4+5+6+ 7
3+4+5+6+7+ 8
4+5+6+7+8+ 9
5+6+7+8+9+10
6+7+8+9+10+11

The sum:

1+2+3
2+3+4
3+4+5

+

1+2+3+4
2+3+4+5
3+4+5+6
4+5+6+7

+

1+2+3+4+5
2+3+4+5+6
3+4+5+6+7
4+5+6+7+8
5+6+7+8+9

equals 6^3

Last edited: Apr 15, 2004
2. Apr 15, 2004

### MathematicalPhysicist

when you draw on a piece of paper a cube you are actually compressing the cube in a two dimension (in a plane).

p.s
i dont get the numbers summations.

3. Apr 15, 2004

### ahrkron

Staff Emeritus
I don't see any relation between your sums and compression of 3D into 2D. They seem to show a property of some partial sums taken from sums that add to cubes.

4. Apr 17, 2004

### Russell E. Rierson

The volume of a 3 dimensional space, "n^3" , is the sum of the elements in a 2 dimensional[square] array, which is the scalar product of two n+k dimensional vectors.

1+2+3 = 6
2+3+4 = 9
3+4+5 = 12

6+9+12 = 27 = 3^3

< 1, 2, 3, 4, 5 >*< 1, 2, 3, 2, 1> =

1*1 + 2*2 + 3*3 + 4*2 + 5*1 = 27 = 3^3

1+2+3+4 = 10
2+3+4+5 = 14
3+4+5+6 = 18
4+5+6+7 = 22

10 + 14 + 18 + 22 = 64 = 4^3

<1,2,3,4,5,6,7>*<1,2,3,4,3,2,1> =

1*1+2*2+3*3+4*4+5*3+6*2+7*1 = 64 = 4^3

5. Apr 18, 2004

### Russell E. Rierson

Three equidistant[comoving] points form an equilateral triangle ABC

Rotate the equilateral triangle to BCA, CAB, it is invariant to ABC

A B C
B C A
C A B

the invariance of rotation for comoving points A,B,C appears to correspond to an array of elements in a 2D[square] matrix. Information is encoded on the surface of space.

According to Hawking, the maximum entropy of a closed region of space cannot exceed 1/4 of the area of the circumscribing surface A/4 .

So information is stored on the 2 dimensional boundary of space analogously to the way a 3D holgram can be encoded on a 2D surface.

6. Apr 20, 2004

### Digit

0D = d0 ; 1
1D = d1 d0 ; 1 2
2D = d1 d0 dd1 dd0 ; 1 2 3 4
3D = d1 d0 dd1 dd0 ddd1 ddd0 ; 1 2 3 4 5 6 7 8

3D contains 2D and 1D and 0D

7. Apr 24, 2004

### MathematicalPhysicist

i feel this thread is way too "developmental" if the moderators know what i mean.

8. Jun 9, 2004

### lvlastermind

Is your quesiton along the lines or combining something like 7x^3 + 3x^2?