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.
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
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/
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.