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Square Cubes

  1. Apr 15, 2004 #1
    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. jcsd
  3. Apr 15, 2004 #2
    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.
     
  4. Apr 15, 2004 #3

    ahrkron

    User Avatar
    Staff Emeritus
    Gold Member

    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.
     
  5. Apr 17, 2004 #4

    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
     
  6. Apr 18, 2004 #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.
     
  7. Apr 20, 2004 #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
     
  8. Apr 24, 2004 #7
    i feel this thread is way too "developmental" if the moderators know what i mean.
    :biggrin:
     
  9. Jun 9, 2004 #8

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