Compress 3D Objects Into 2 Dimensions?

In summary: If so, then the answer is no. Objects can only be compressed into two dimensions if they are drawn on a two dimensional surface.
  • #1
Russell E. Rierson
384
0
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
 
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  • #2
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.
 
  • #3
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
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
 
  • #5
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
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
i feel this thread is way too "developmental" if the moderators know what i mean.
:biggrin:
 
  • #8
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?
 

1. What is the purpose of compressing 3D objects into 2 dimensions?

The purpose of compressing 3D objects into 2 dimensions is to reduce the amount of data needed to represent the object, making it easier to store and manipulate. It also allows for faster processing and rendering of the object.

2. How is the compression process carried out?

The compression process involves converting the 3D object into a 2D representation, typically through techniques such as projection or slicing. The resulting 2D image or data can then be stored and transmitted more efficiently.

3. What are the benefits of compressing 3D objects into 2 dimensions?

Aside from reducing storage and processing requirements, compressing 3D objects into 2 dimensions can also improve visualization and analysis of the object. It can also make it easier to share and distribute the object with others.

4. Are there any drawbacks to compressing 3D objects into 2 dimensions?

One potential drawback is that some information may be lost in the compression process, resulting in a less accurate representation of the 3D object. Additionally, certain features of the object, such as texture and lighting, may be affected by the compression.

5. What are some applications of compressed 3D objects?

Compressed 3D objects have a wide range of applications, including in video games, virtual and augmented reality, medical imaging, and architectural design. They are also commonly used in data visualization and scientific simulations.

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