"Coordinates of Each Shape"

[luisegarduno]

Reference - see Google Doc

[luisegarduno]

General Notes

• All lines are equal to 1
• All vertices, except those at the origin, will be in the positive direction, lying within the first "sedecant" (+, +, +, +)
  • I (@Levi-B4) derived sedecant from sedecim (16) and quadrant/octant.
• Circles will have a focus at (0, 0, 0, 0)
• When creating the program, the center of every shape will be put at (0, 0, 0, 0) This is done by offsetting in the negative direction by subtracting: (x_max/2 , y_max/2 , z_max/2 , w_max/2)

[Levi-B4]

Circles

General Notes

Coordinates for circles in each dimension will be found using the equation of a circle (x − h)² + (y − k)² = r² and extending this to other dimensions. The centroids will be at (0,0,0,0) and the radii will be 1, the same as the side length of other shapes. The equation of a circle per dimension will be

 (a₁ − c₁)² + (a₂ − c₂)² + … + (a_d − c_d)²

where:

• a = axis (a₁ = x, a₂ = y)
• c = centroid of axis
• d = dimension

1D Line

• (x − h)² = r²
  • x = ±1
  • (-1,0,0,0)
  • (1,0,0,0)

2D Circle

• (x − h)² + (y − k)² = r²
  • x² + y² = 1

3D Sphere

• (x − h)² + (y − k)² + (z - m)² = r²
  • x² + y² + z² = 1

4D 3-Sphere

• (x − h)² + (y − k)² + (z - m)² + (w - n)² = r²
  • x² + y² + z² + w² = 1

[Levi-B4]

Squares

General Notes

Coordinates for squares in all dimensions can be found by extruding the square of the previous dimension by the side length in the direction of the new dimension. You can represent these coordinates with binary counting. Each bit corresponds to an axis so a bit is added for each dimension. 0000 = xyzw = (0,0,0,0)

• 1D: 0-1 x-x
• 2D: 00-11 xy-xy
• 3D: 000-111 xyz-xyz
• 4D: 0000-1111 xyzw-xyzw

1D Line

  (0,0,0,0)   (1,0,0,0)

2D Square

  (0,0,0,0)   (0,1,0,0)   (1,0,0,0)   (1,1,0,0)

3D Cube

  (0,0,0,0)   (0,0,1,0)   (0,1,0,0)   (0,1,1,0)   (1,0,0,0)   (1,0,1,0)   (1,1,0,0)   (1,1,1,0)

4D Tesseract

  (0,0,0,0)   (0,0,0,1)   (0,0,1,0)   (0,0,1,1)   (0,1,0,0)   (0,1,0,1)   (0,1,1,0)   (0,1,1,1)   (1,0,0,0)   (1,0,0,1)   (1,0,1,0)   (1,0,1,1)   (1,1,0,0)   (1,1,0,1)   (1,1,1,0)   (1,1,1,1)

[Levi-B4]

Triangle

I still need to find a simple way to scale dimensions, It is close but not yet complete. The next comment will have the dimensional pattern. For now I will note that with each dimension of a triangle, including the 0th dimension, a new vertex is added. For example a four dimensional triangle has 5 vertices.

0D Point

• [0,0,0,0]

  1D Line

• [1,0,0,0]

  2D Triangle

• [ $1 \over 2$ , $\sqrt{3} \over 2$ , 0 , 0 ]

  3D Tetrahedron

• [ $1 \over 2$ , $\sqrt{3} \over 6$ , $\sqrt{2} \over 3$ , 0 ]

  4D 5-Cell

• [? , ? , ? , ? ]