# Some Geometric Ideas

• imagine a single point in 3-d space
• moving a point will trace out a line (two end points; one line)
• moving a line will trace out a square (four points; four lines)
• moving a square will trace out a cube (eight points, twelve lines)
• moving a cube will trace out a hyper-cube
Object
Number
Object Name dimensions vertices
(points)
edges
(lines)
faces
(planes)
cubes hypercubes   dimensions vertices
(points)
edges
(lines)
faces
(plains)
cubes
Progression of a dot:             Unfolded:
1 point 0 1 - - - -   - - - - -
2 line 1 2 1 - - - point 0 2/1 - - -
3 square 2 4 4 1 - - line segments/line 1 5/2 4/1 - -
4 cube 3 8 12 6 1 - flat cross 2 14 19 (6) -
5 hypercube (Tesseract) 4 16 ? 32 ?  24 ? 8 ? 1 8 cubes in a cross 3 36 44 40/29 8
Other Stuff:
3 sided pyramid (Tetrahedron) 3 4 6 4 - - 4 triangles 2 6 9 (4) -
4 sided pyramid 3 5 8 5 - - 4 triangles, 1 square 2 8 12 (5) -
5 sided pyramid 3 6 10 6 - - 5 triangles, 1 pentagon 2 10  15 (6) -
Octahedron 3 6 12 8/(9) - - 8 triangles 2 10  17 (8) -
Dodecahedron 3 20 30 12 - - 12 triangles 2 38  49  (12) -
Icosahedron 3 12 30 20 - - 20 triangles 2 22 31 (20) -
• The Euler Formula (below) is always true for 3-d objects consisting of straight lines:
Edges = Vertices + Faces + 2
• The yellow diagonal extrapolation suggests 8 cubes may be created when a 3-d cube is moved through a fourth (time?) dimension
• Thought experiment: moving a cube from A to B in a 3-dimension space maps out lines between two cubes (one beginning; one ending) with connecting lines between the corners. A different visualization pictures a smaller cube inside a larger one with lines connecting the closest corners. When drawn out with pencil and paper you can see:
• 16 points where lines connect. The green vertical extrapolation suggests that this thought may be correct for four dimensions.
• 32 lines ((2 x 12) + 8 new ones) but this may not be valid for 4 dimensions.
• 24 planes ((2 x 6) + 12 new ones) but this may not be valid for 4 dimensions
• all 8 cubes if you search long enough.
• Observations:
• The following formula is consistent for objects 1-5:
Vertices = 2 ^ Dimensions
• Here's something I just noticed for objects 2-5:
Dimensions x Vertices / Edges = 2

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Neil Rieck
Kitchener - Waterloo - Cambridge, Ontario, Canada.