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Outside the Box Geometry

Fun, challenging geometry that will shake up how you think and problem solve!

The Beautiful

The image on the left shows how hexagons can cover the plane without gaps or overlaps.

Can copies of the purple region on the right be arranged to cover the plane without gaps or overlaps?

Note: The purple region was created by starting with a circle, and removing a part corresponding to two quarter-circles with the same radius as the original circle.

The most important skill in geometry is creativity, not memorization. That’s why this Exploration includes many beautiful geometry topics that are unlike anything you’d typically see in a school classroom!

All of the questions in this quiz are about a research topic that you’ll investigate in depth later in this exploration:

Tessellations (the Math of Tiling the Plane)
M.C. Escher's art isn't just beautiful - it's also extremely mathematical!

3D Polyhedra and Euler's Facets Formula
We'll dive into three dimensional polyhedra, including “Euler’s formula” which unites geometry and graph theory.

The Mathematics of Folding Paper
In the art of origami (paper folding), we'll learn to understand the crease patterns used to create origami sculptures.

Irregular Polygons (Pegboards and Museum Guard Puzzles)
Most real-world polygons are irregular, and we'll see problems from designing on pegboards to guarding an art museum.

A rhombicuboctahedron is a 3D solid that has 18 square faces and 8 triangular faces. How many toothpicks were used to the model rhombicuboctahedron pictured above on the right?

Hint: You can solve this puzzle without counting all of the toothpicks by noting that each edge is shared by exactly two faces.

To make origami, you fold a square twice (in succession). Which of the options below is a possible crease pattern when the square is unfolded?

Later in this exploration, we'll see how to crease paper to form complex origami like this crane:

What is the area of the irregular orange polygon in this grid of equally spaced dots?

While you can solve this problem in many ways, we'll later explore Pick’s theorem: a clever trick for finding the area of irregular polygons drawn on pegboards.

The irregular purple polygon above is the floor plan of a gallery, and an example is shown of what could be seen by a single guard in a given location.

Your job is to position some number of unmoving guards - who cannot see through walls - such that every location in the gallery is in view of one of the guards.

What is the fewest number of guards that you could use?

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