by Ed Pegg Jr and George Hart

Regular Pentagon and Center

Make a regular pentagon of any size, and a node at the
exact center. You may use any manner of connecting struts.

K4 (four connected nodes)

See http://www.zometool.com/manual/tetra/tetrachart.gif
for a complete list.

K5 (five connected nodes)

Three B1, Three Y1,
One Y2, and Three R1.

Two B1, One B2,
Three Y2, Three R1,
and One R2.

Three B1, One B2,
Five Y1, and One R2.

Two B2, Three Y2,
One Y3, One R1,
Two R2, and Three R3.

K6 (six connected nodes)

Four B1, Three B2,
Six Y1, and Two R2.

Five B2, Two B3,
Four Y3, and Four R1.

K7 (seven connected nodes)

Six B1, Three B2,
Six Y1, and Six G1.

Overhand Knot

Two B1, Two B2,
One Y1, and One R2.

Three B3, and Three Y3
(slight bending)

Three B3, and Three G2

Graph Construction (each node has
three struts)

One of each: B1 B2 B3 Y1
Y2 Y3 R1 R2 R3

Two B1,
Three B2, Three
Y1,
and One R2.

Two B2,
One B3, Three
Y2, One R1,
and Two
R2.

One B1,
One B2, Two Y1,
Three Y3, One
R2,
and One R3.

One B1,
Two B2, One B3,
Two Y2, One Y3,
One R1, and One
R2.

One B2,
One B3, Two Y2,
Two Y3, One R1,
One
R2, and One
R3.

Construction Problems from http://www.georgehart.com/

Remember, in each polyhedron, all the faces are identical. Be sure your faces are planar, e.g., four vertices do not form a kite if they are not in the same plane. Also, we do not allow adjacent faces to be coplanar. For example you can’t just add the long diagonals to the kites and call that a solution to B.

Most puzzles should be RIGID structures. A figure is RIGID if every node has 3 or more struts, and every strut has 2 nodes. All of the 4 node tetrahedra in the link above are rigid.

At the page for the 34 forms of convex heptahedra, solutions have been found for the blue colored cells in the table.

I also plan to look at knots, such as the ones at http://www.cs.ubc.ca/nest/imager/contributions/scharein/cal/inlines/BackCover.jpg

My puzzle ideas are thus involved in Graph Theory and Knot Theory, primarily. Any other ideas?

I’m tempted to use Chime at http://www.mdli.com/cgi/dynamic/welcome.html,
but it seems that an actual Zome program would be more interesting.