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.
“Tessellations. You know, shapes that repeat themselves to fill up a plane.”
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?”
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!”
“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.