Some Geometric Ideas

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.