by wendy » Wed Sep 06, 2017 9:12 am
While the 16-cell can be derived from the teseract by alternating vertices, there is also a derivation of the 16-cell by way of thirds of the 24-choron. The three 16-cells then sit together as
xoo3ooo3oxoAoox
This can be rended to a six-dimensional lace prism by the addition of &#x at the end, which would cause a triangle of these "16-cell antiprisms", each pair deriving a different penteract they come from.
Prisma is derived from a greek word 'to cut'. In three dimensions, the only prisms derived are those where a line is one of the factors. But in four dimensions, one can derive prisms where there is no opposite face. The bi-triangular pyramid, or triangle-duoprism, sitting on one face, the upmost surtope is not a face, but the triangle margin between two faces. Not withstanding, this prism translates to on the w,x space, a triangle, and on the y,z space, a triangle.
While prisma derives from 'offcut', such as one might strike from a rod of a given section, a different way of making prisms is to roll a layer of dough, and cut shapes using a pastry or cookie cutter. The first cut is then to roll the layer of pastry, the second is to use the cookie cutter to make the shapes.
The trouble with science, is that the variation of word use from art to art, causes many of the confusions to students, and one gets obnoxious comments on the likes of Wikipedia, where people wrongly suppose that weight is a force in 'weights and measures' (one art), where in the matter of mechanics it is a force. One only has to look at the talk-pages of things like pound, troy pound, etc to see this.
To this end, it is best to study what the word in common use means, and to make the central meaning of the word in the current art take the same. For example, physics does have a meaning corresponding to 'weight' in common use, but because they misappropriated weight to something else, they had to invent a new word. So we have 'mass', which is elsewhere something that is measured with a ruler (Mass und Gewichte).
Reading the old geometries, such as that of D M Y Sommerville or H S M Coxeter, does not help here. Coxeter uses 'cell' in the sense of face, and sometimes as a three-dimensional element. Sommerville uses 'hedroid' for a choron.
The trick we use is to suppose that the words in common use are relative to solid space, so that a plane is represented by one equal sign (ie z=... ), and that a margin is by two equal signs, eg y=., z=. New names are then created for the meanings of space of specific dimension, this derived by a back-forming of the word 'polyhedron' = poly (many + closure) + hedr (2D) + on (patch). Using various different elements it is possible to invent a word to describe E6, as a horo (E) + ect (6d) + ix (fabric).
The products are according to construction, not outcome. A prism is supposed to be cut from shapes of its bases in orthogonal axies. So a pentagon prism can be struck from a pentagon-bar (ie x,y,z -> pentagon, z -> pentagon, height), or by pressing a pentagonal cutter against a slab, ie x,y,z -. x,y, height -> pentagon, height. There are something like five known products, all of which reduces one or more properties of the bases, to algebraic forms that multiply to the property of the result.
With prisms, the dual is tegum. Cut and cover, so to speak. For all of the various uses of prism, there is a matching construction to the dual, which is a tegum. The antiprism, as constructed by the draught of duals, provides us with interesting insights of the figures described. The tegum-product of several anti-prisms is itself an antiprism, being the pyramid product of any base of each of the elements.
The terminology of ordinary geometry does not scale well. A plane in 3D, is ++=, that is a 2-dimensional thing given by a single equal-sign (z=0). But the reality is, that in four dimensions, there is no such thing. It requires four items, ie +++= or ++==. One can scale it either way. But the students are coming into the class with in mind, the plane is a dividing space, ie z=0, gives ~=. The ~ is then filled with as many + as needed. But to invent a new word for this meaning is to create some sort of confusion in the reader that a plane qua ++== can be woven as a cloth, but you can't weave +++=.
And this is why i spend a very large measure of time restructuring the words to match the expectations of the reader, rather than 'first impressions' of the author. It's not a popular path, but I find it quite necessary, because too much exotic happens up there.
The dream you dream alone is only a dream
the dream we dream together is reality.