HYPERCUBE & PENTATOPE #hypercube #pentatope

February 27, 2025

HYPERCUBE:

A hypercube is 4th dimensional analog of a cube . Hypercube is essentially a cube extended into a fourth dimension.

A hypercube has 8 cubes(3D) as faces, 16 vertices(0 D), 32 edges(1D),24 faces which are 2D squares.

{ Here D indicates the dimension}

In this hypercube all edges meet at right angles.

Basically if we take a 3D cube , a 3D cube has height, width, breath and area whereas the 4D cube i.e. a hypercube has all the above height, width, breath and area thrice i.e. inner, outer and total.

That means hypercube has inner height, outer height and total height, inner width, outer width and total width just like the other two breath and area are also measured for inner, outer and total . Hence totally a hypercube has 12 properties.

PENTATOPE :

A pentatope is a fourth dimensional geometrical shape and pentatope is a fourth dimensional analogue of tetrahedron. A pentatope is also called a 5-cell because it consists of 5 vertices.

A pentatope contains 5 vertices, 10 edges, 10 triangles and 5 tetrahedrons.

In the following figure I mension a pentatope with 5 vertices named as A,B,C,D,E , 10 edges named as the numbers from 1 to 10 and 10 triangles by Roman letters I to X for convenience.