Kitchen floor ceramic Penrose Tessellation handmade from tiles Gallery: Tessellations in Real Materials like wood, clay, and chalk Tamas Szigeti's spherical tessellation in carved wood

Real materials 17:
5-Cat Dodecahedron Tessellation

12 sided 3D dodecahedron tessellation made from paper

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.

paper 3D geometry dodecahedron tessellation made from 12 paper pentagons

To figure that out, just lightly write a letter on each side of your pentagons like I've done in the picture at left.. Write "A, B, C, D, E" , one letter per pentagon side, in (clockwise or counterclockwise) order. Next, press your pentagons together, flat on a table, like the picture at left. See how Side A on one pentagon matches up with A on the pentagon next to it. Similarly, Side B fits with another pentagon's Side 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, instead of treating a single pentagon as the basic unit, try using a group of pentagons as the basic unit. Try grouping the 12 pentagons into clusters, and let each repeating shape roam not-just-around-one-tile, but take up several adjacent tiles: 6 groups with 2 pentagons each, 4 groups of 3 pentagons each, 3 groups containing 4 pentagons each, and 2 groups with 6 pentagons in each group.

...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!

paper 3D geometry dodecahedron tessellation made from 12 paper pentagons