Naked Hypercube

Hypercube: Beyond the Third Dimension

Logically there is no limit to the number of spatial dimensions. Current thinking suggests that our universe exists in at least ten dimensions. Because our experiences are largely confined to three dimensions, it is difficult to visualize more than three axes which meet at right angles. Nonetheless, properties of higher-dimensional objects are easy to access by extending relationships between other dimensions.

Zero. A point has no length, no breadth, no width.	One. Stretching a point in any direction, say along the x-axis, creates a one-dimensional line segment.
Two. A one-dimensional line segment can be stretched along an orthogonal axis (in the y-direction) to make a 2D square.
Three. If a 2D square in the x-y plane is stretched in the z direction, a 3D cube may be formed.
Four. Stretching a cube along the w-axis (perpendicular to each of the x-, y-, and z-axes) can create a 4D hypercube.

Geometry in 4D

The 4D hypercube is also called a tesseract, an 8-cell, or an octachoron.

Its attributes can discovered by extending relationships between lower dimensions, as shown here.

	0d	1d	2d	3d	4d
v	1	2	4	8	16
e		1	4	12	32
f			1	6	24
c				1	8
t					1

Rows indicate attributes:

v = number of vertices
e = number of edges
f = number of faces
c = number of cubes
t = number of tesseracts

Columns indicate dimension of orthotope.

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