Have you ever played with shapes and patterns? That's what geometry is all about! It's the study of lines, angles, shapes, and space. From drawing cool designs to building structures, geometry is an important part of our world.

The overall objective of this project was to explore which shapes can (or cannot) tile a rectangular grid or infinite plane (in one or multiple ways) and understand why.

Discover which shapes can tile an infinite plane and why! Experiment with pentominoes, heptiamonds, wheelbarrow, and kite #n# dart pairs to find the most efficient shape for tiling.

Project Title: Tiling with Shapes and Tessellations in Nature

Objectives/Goals

The overall objective of this project was to explore which shapes can (or
cannot) tile a rectangular grid or
infinite plane (in one or multiple ways) and understand why.
Methods/Materials
We discovered many different types of amazing shapes that have been used in
tiling in man-made objects
such as puzzles, art, and architecture, and tessellations in nature such as
crystals and honeycombs. We
decided to experiment with the following shapes: Pentominoes, Heptiamonds,
Wheelbarrow, Kite #n#
Dart Pair, and Regular Convex Polygons (including Triangles, Squares, and
Hexagons).
We built pentominoes from legos, heptiamonds from pattern blocks; wheelbarrow
and kite #n# dart from
tagboard, and different types of (triangular, square, and hexagonal) honeycombs
with manipulatives. We
experimented tiling appropriate rectangular grids and planes using these shapes.
We used a symmetry principal to reduce the number of tiling problems for
pentominoes.
We counted the number of sides used in building honeycombs and noticed patterns
for which we derived
formulas for the amount of wax used.

Results

We found that we can tile some rectangular grids (8x8 with a square removed
anywhere, 3x20, 4x15,
5x12, and 6x10) with twelve pentominoes. Additionally, we discovered that all
but one of the 24
heptiamonds could tile a plane individually. We invented interesting,
non-trivial ways of tiling a plane
using both wheelbarrow and kite #n#dart pairs. Finally, we used manipulatives,
algebra, and geometry to
prove that hexagons are the most efficient shape (consume the least wax to
create the same amount of
area) from all regular convex shapes to build a honeycomb.

Conclusions/Discussion

We discovered that although many different types of convex and non-convex shapes
(e.g. pentominoes,
wheelbarrow) and several interesting combinations of them (e.g. kite n#dart) can
be used to tile a plane in
interesting ways, there are many simple shapes that cannot be used to tile the
plane (e.g. a V-shaped
heptiamond or a pentagon). Although we proved that hexagons use the least amount
of wax in comparison
to squares or triangles, we did not prove that they are better than irregular or
multiple shapes to hold
honey (we discovered that the general proof was given only five years ago in
1999!).

Summary Statement

In this project we explored which shapes can or cannot tile an infinite plane or
rectangular grid, explained
why, and applied our findings to nature and society.



Tiling with Shapes and Tessellations

Which shapes can tile an infinite plane?

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Math is all about numbers, patterns, and solving problems. Computer science is about using math and technology to create computer programs and solve real-world problems. Together, they help us understand how things work, how to solve problems, and how to use technology to make our lives better!

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Tiling with Shapes and Tessellations

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