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used with permission

Fulcrum Design Logo
mathematically-themed sculpture puzzles that you assemble yourself
  designed by Tom Longtin

figure 1 My puzzles are based on the geometry of hexagons, a form appearing frequently in nature as honeycombs, the markings on a turtle's shell, and snowflakes.

As in nature, the puzzles contain layers of complexity within their outwardly simple forms.  
figure 1. the puzzles are based on the geometry of hexagons

figure 2
Hexagons may be evenly divided by bisecting the middle and sectioning off two exterior parts (figure 2).
figure 2. divided hexagons

figure 3 After reassembling the hexagons, the sectioned paths make unique closed pathways when the 2D plane is rolled into a tube, and then into a torus
figure 3.  sectioned pathways

figure 4
Contiguous unique areas on the torus surface are defined by these paths (figure 4). 
figure 4

figure 5
Filling the contiguous areas with color highlights the continuity of the defined area (figure 5).
figure 5

figure 6
Both the Trefoil Knot and Step-Star are topologically torus shapes, and can be assembled inside out (although the handedness of the Trefoil Knot will reverse).  Furthermore, both puzzles are subject to the seven-color theorem for torus shapes; click here for more information about color research by Stan Tenen.
figure 6. a torus
used with permission

figure 7
Challenge I:

Each piece has a distinctive field within it (see figure 7).  How many possible ways may the puzzle be assembled?
figure 7.  the puzzle pieces are based on the regular hexagon, re-interpreted as a six-sided form with 90 degree angles

Challenge II:

1) After assembly, paint or mark continuous fields, continuing the color around the 90 degree edge, but stopping at the field edges (figure 8). 

2) Disassemble the puzzle.

3) Re-assemble, color-matching continuous fields.

4) How many possible ways are there to re-assemble the puzzle with continuous color fields?  With dis-continuous color fields?
figure 8

figure 8
To pose a new challenge, or ask questions, please contact: 

©2009 Tom Longtin