A printer-friendly (condensed) version of this page is available here.

back to Regular Polytopes Project

What is the hypervolume of an n-dimensional hypercube?

Hint: What are the hypervolumes of a line segment, a square, and a cube?

back to the project

How many vertices does an n-dimensional hypercube have?

Hint: How does the number of vertices change in constructing an n-dimensional hypercube from an (n – 1)-dimensional hypercube?

back to the project

What is the number of edges contained in an n-dimensional hypercube in terms of the numbers of vertices and edges in an (n – 1)-dimensional hypercube?

Hint: How does the number of edges change in constructing an n-dimensional hypercube from an (n – 1)-dimensional hypercube?

back to the project

What is b_{n, m} for hypercubes in terms of previous entries?

Hint: In constructing an n-dimensional hypercube from an (n – 1)-dimensional hypercube, each (m – 1)-dimensional hypercube in the (n – 1)-dimensional hypercube contributes an m-dimensional hypercube in the n-dimensional hypercube.

back to the project

What is b_{n, m} for hypercubes?

Hint: The previous recurrence for b_{n, m} must be solved explicitly. First find b_{n, 1} by repeatedly substituting the recurrence relation into itself until a closed-form expression is obtained. An expression for b_{n, 2} can be found similarly, etc.

back to the project

What is the surface hyperarea of an n-cube?

Hint: How many (n – 1)-cubes are there in the n-cube, and what is the hypervolume of each?

back to the project

How many vertices does an n-simplex have?

Hint: The natural generalization is the right one.

back to the project

What is b_{n, m} for simplices?

Hint: Notice that every set of three vertices in an n-simplex forms a triangle, every set of four forms a tetrahedron, etc. Binomial coefficients count the number of these objects.

back to the project

What is the height of a regular n-simplex?

Hint: The hypervolume of an n-dimensional pyramid is h · A / n, where h is the height of the pyramid and A is the hyperarea of the base. What is the hypervolume of an n-simplex in terms of the apothem a_n and its (n – 1)-dimensional surface hyperarea? By equating the two, one can find the height in terms of n and a_n.

back to the project

What is the hypervolume of a regular n-simplex in terms of s?

Hint: Find h_n by considering right triangles.

back to the project

back to Regular Polytopes Project