# Powertope (EntityClass, 3)

### From Hi.gher. Space

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 A^{P} 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.