Real materials 17:
5-Cat Dodecahedron Tessellation

This paper or cardboard 12-sided 3D tessellation by Seth B. is easy to make, and an ideal 3D geometry project.

You'll need scissors, glue, a printer, and paper. Just print this PDF file of a single pentagon with cat outlines. Print it 12 times, color the cats, and glue the 12 pentagons together as shown in the top picture on this page.

Hint: separate the 12 pentagons into two groups. Put one in the center of each group and glue its edges to 5 pentagons around its edges, so it looks like a flat flower with five petals. (See the middle picture on this page.) Wait until they dry, then glue the petals together and let them dry. At this point, you'll have two cup shapes of 6 pentagons each. Finally, glue the two cup-shaped halves of the dodecahedron together.

3D tessellation really isn't difficult. For a "dodecahedron", the only trick is to know which pentagon sides will touch one another when the "dodecahedron" is assembled.

To figure that out, just lightly write a letter on each side of a pentagon. Write "A, B, C, D, E" , one letter per side, in (clockwise or counterclockwise) order. Side A matches up with A; side B fits with C (and vice versa), and side D fits with E (and vice versa). In the cat tessellation at the top of this page, the tall gray cat is on side "A" of each pentagon-- see that every tall gray cat matches up with another tall gray cat on another pentagons.

To make your own dodecahedron tessellation, cut out 2 pentagons on tracing paper. Letter their sides, and then match them up, one pair of sides at a time. Draw a squiggly borderline between the paired sides. Why're we using tracing paper? Because we have to be sure the borders along both pentagons' A-sides are identical, and both Bs, Cs, Ds, and Es.

For extra fun, treat a group of pentagons as the smallest repeating unit in the tessellation, just as you earlier used single pentagons as the smallest repeating unit. Try grouping the 12 pentagons into

• 6 groups with 2 pentagons each,
• 4 groups of 3,
• 3 groups of 4, and
• 2 groups of 6.

...then match up their sides. Beware: you can't use the old A-B-C-D-E formula that we use for single pentagons. We need a unique formula for each grouping, to figure out which sides will match up. Good luck!