Alexandria Jones hated using store-bought wrapping paper at Christmas. She tried to wrap each present as a hand-crafted work of art.
Last year, she did mini-scenes with plastic figures building cotton snowmen or skating on aluminum-foil ponds — and, for her brother Leonhard’s gift, her favorite creation: toy dinosaurs having a snowball fight. But those 3-D scenes got knocked about under the Christmas tree.
This year, she decided, she would wrap the packages flat. But then, how could she make them special?
Introducing Alex’s Mother
Maria Jones stood at the kitchen table, shaping wreath cookies. She watched as her daughter tore up her fifth drawing in seven minutes. Alex grabbed a clean sheet of paper and began scribbling anew.
“Have you considered tessellations?” Maria asked.
“What?”
“Tessellations. You know, shapes that repeat themselves to fill up a plane.”
“Oh, I see,” Alex said. “Like squares on a checkerboard.”
Her mother nodded. “Squares are probably the simplest tessellating shape,” she said. “Why don’t you see how many others you can find?”
Alex sketched a checkerboard design, then started a new pattern with slanted lines.
“Diamonds will tessellate,” she said. “And what if I cut each diamond in half?”
“Hey — triangles tessellate, too!”
Don’t Jump to Conclusions
“Be careful with generalizations,” Maria said. “Those look like isosceles triangles. What about scalene ones?”
“Scalene. That means an irregular triangle, right? Let’s see…” Alex flipped her paper over and began another sketch.
She scratched it out, bit her lip, and tried again. Finally, she put her pencil down and looked up at her mother. “I think it works, but how can I be sure I’m drawing the same triangle over and over?”
Maria smiled. “Well, when I’m making a quilt, I—”
“Use a template. Of course!”
Alex jumped up and ran to the cupboard. She tore a flap off a half-empty cereal box and cut it into a triangle.
“This works great!” she said. “And look: If I squiggle the short side of the triangle, I can turn it into a rocket.”
If you want to see Alex’s tessellation designs, download this page from my old newsletter: Tessellation Wrapping Paper, pdf 70KB.
To wrap a present, Alex would need a big sheet of paper. She grabbed her pencil sketch and headed for her room. But she turned around in the doorway.
“Thanks, Mom,” she said. “Tessellations are a great idea!”
Try It for Yourself
Alex continued playing around with tessellations, and she found several more shapes that would tessellate.
How many can you find?
Edited to Add
Answers to this and other Alexandria Jones puzzles are now posted:
To Be Continued…
Read all the posts from the November/December 1998 issue of my Mathematical Adventures of Alexandria Jones newsletter.
Denise Gaskins' Let's Play Math 
Now I wonder if Niner has wrapped Christmas gifts for Smokey Joe (Red’s name for him) and me with tessellations. I guess she will have to tell me because I didn’t know the word before now.
No, she hasn’t. Her creativity has been mostly with bows and ribbons. I got the idea for this post from the 3-D scenes I used to create on presents. Do you remember the Kremlin model? I’m sure I did snowmen and a frozen lake sometime, although I don’t think I had little ice skaters for it.
Nowadays, we are always running at the last minute, so we haven’t had time to do this sort of thing. But it would be fun to try it again. I wonder if we can get ahead of the calendar this year…?