Compress 3D Objects Into 2 Dimensions?

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Discussion Overview

The discussion centers around the possibility of compressing a 3D object into 2 dimensions, exploring mathematical representations and theoretical implications. Participants examine various examples, including summations and geometric interpretations, while questioning the relationship between these concepts and dimensional compression.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • Some participants propose that drawing a cube on paper represents a compression of the cube into two dimensions, although the mathematical summations provided are unclear to others.
  • Others argue that the sums presented do not directly relate to the concept of compressing 3D into 2D, suggesting they illustrate properties of partial sums related to cubes.
  • A participant discusses the relationship between the volume of a 3D space and the sum of elements in a 2D array, presenting scalar products of vectors as a means to demonstrate this connection.
  • Another viewpoint introduces the idea of information being encoded on the surface of space, referencing Hawking's principle regarding entropy and the maximum information capacity of a closed region.
  • One participant outlines a dimensional hierarchy, stating that 3D contains 2D, 1D, and 0D, suggesting a nested structure of dimensions.
  • A later reply questions whether the original inquiry relates to combining polynomial expressions, indicating a potential misunderstanding of the initial question.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between the mathematical examples given and the concept of dimensional compression. No consensus is reached, and multiple competing interpretations remain present throughout the discussion.

Contextual Notes

Some mathematical steps and assumptions are not fully resolved, leading to ambiguity in the connections drawn between the examples and the main question of dimensional compression.

PFanalog57
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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:
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Russell E. Rierson said:
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
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 don't get the numbers summations.
 
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.
 
ahrkron said:
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.


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
 
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.
 
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
 
i feel this thread is way too "developmental" if the moderators know what i mean.
:biggrin:
 
Russell E. Rierson said:
Is it possible to compress a 3D object into 2 dimensions?


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

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