Extruding from three-dimensions to four.

Higher-dimensional geometry (previously "Polyshapes").

Extruding from three-dimensions to four.

Postby pat » Fri Sep 08, 2006 11:06 pm

Take a solid three-dimensional object. Facetize its surface. Now, extrude each facet one unit in a direction perpedicular to the three-space in which the object sits.

The analogous thing down a dimension is to take a solid, planar shape. Polygonalize its boundary. Extrude the boundary one unit perpendicular to the plane of the object.

If you duplicate the solid polygon on both top and bottom, you end up enclosing a three-dimensional volume. By enclosing, I mean that the shape divides three-space into two regions, a bounded interior and an unbounded exterior.

If you did the same thing in four dimensions, you will have enclosed a four-dimensional volume.

My problem is that I don't want to have to deal with the solid three-dimensional object. I only want to deal with its facetized surface. If I start from a point, taper each 2-facet of my 3-object up to full size as I extrude in the w-direction, then taper back to a point as I extrude further, have I enclosed a four-volume? Effectively, this is turning each facet into a bipyramid and all bipyramids share the same "upper" vertex and the same "lower" vertex.

The dimensional analogy says yes. My intuition says yes. All of the vectors that I've tried to cast "through" the object hit it. Am I missing something? or am I right?
pat
Tetronian
 
Posts: 563
Joined: Tue Dec 02, 2003 5:30 pm
Location: Minneapolis, MN

Postby wendy » Sat Sep 09, 2006 7:02 am

What you have is a simple tegum product of a figure in xyz and the line in w. The surface does indeed close.

Consider. ( -> give also)

hedra in 3d -> chora in 4d
edges in 3d -> hedra in 4d
points in 3d -> lines in 4d

There is also a new apex created at each end.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia


Return to Other Geometry

Who is online

Users browsing this forum: No registered users and 9 guests