A Measure Polytope can be described as the 4-space member of a class of regular rectilinear prisms.  In general these prisms are formed by moving the prism of the next lower space in a direction perpendicular to that space a unit of measure equal to all previous movements.  The table below describes the characteristics of the rectilinear prisms in the first four dimensions.
Image
V
e
r
t
E
d
g
e
F
a
c
e
V
o
l
u
m
e
H
y
p
e
r
v
o
l
Description
Point
1
Beginning with 0 space, all figures may be represented by a point
Line
2
1
In a 1 space, moving the point along the x axis (perpendicular to the 0 space?) a line is formed.  This prism has 2 vertices and a single edge.
Square
4
4
1
In a 2 space, moving the line along the y axis (perpendicular to the x axis) a square is formed. This prism has 4 vertices, and is enclosed by 4 edges. face.
Cube
8
12
6
1
In a 3 space, moving the square along the z axis a (perpendicular to the xy plane) a cube is formed. This prism has 8 vertices, 12 edges, and is enclosed by 6 square faces.
Hyper Cube
16
32
24
8
1
In a 4 space, moving the cube along the q axis (perpendicular to the xyz space) a hypercube is formed. This prism has 16 vertices, 32 edges, 24 faces, and is enclosed by 8 cubes.
This progression may be extended into as may spaces as the viewer cares to.  For example the number of vertices of 5 space Measure Polytope is 2 times the number of vertices of the figure in the next lower space (2*16=32).  The number of edges is 2 times the number of edges plus the number of vertices from the figure in the next lower space (2*32+16=80).  The number of faces is 2 times the number of faces plus the number of edges in the figure in the next lower space (2*24+32=80).  The number of volumes is 2 times the number of volumes plus the number of faces of the figure in the next lower space (2*8+24=40).  The number of hyper-volumes is 2 times the number of hypervolumes plus the number of volumes from the figure in the next lower space (2*1+8=10).
HyperspaceSimplex
last revised: 9/26/02