# Powertope (EntityClass, 3)

A powertope is a shape formed from a base shape, A, and an exponent shape, P. They can be written as "the P of A". If P is n-dimensional, then AP is the brick product P applied to n repetitions of A.

Powertopes are systematically named as "the A P-triate", with A changed to its adjectival form, any final "n" in P being changed to "l" and any final "te" in "-ate" being removed.

P cannot just be any shape; it must have brick symmetry, and therefore can be represented as the convex hull of a number of irregular hypercubes. This representation must be carried out in order to calculate the shape of a powertope.

All non-trivial powertopes exist only in four dimensions or higher; the simplest non-trivial powertope being the four-dimensional triangular octagoltriate, and the simplest non-trivial usable exponent being the octagon, thus producing the octagoltriates. In each dimension greater than one, there are an infinite number of usable exponents, providing a wide variety of powertopes.

Powertopes can also be called triates, from the naming. This word is derived from the fact that the hyper index of exponentiation is 3, giving tri. Similarly, Cartesian products can also be called diates. There are also tetrates and higher, though these series tend to grow to such extremely large numbers of hypercells and dimensions that their research, even if computer-aided, becomes very impractical; restrictions on the equivalent to the shape P also increase dramatically the further you go up the sequence, with the limit that P is restricted to simply be a hypercube itself as the hyper index increases towards infinity.