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# Beautiful Geometry

## Escape the ordinary by taking an adventure through these beautiful topics.

Are you ready to start loving geometry? This course is here to guide you through some of the magic of geometry, revealing the thought processes that lead to clever solutions to beautiful geometry problems.

By the end of this course, you'll have explored polyominoes, tessellations, origami folding, art gallery problems, and lattice polygons.

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1. 1

### Introduction

Sample the beautiful side of geometry.

1. #### Infinite Areas

How can you fit infinitely many shapes in a finite space?

2. #### Polyomino Tiling

Get your first taste of beautiful geometry by exploring these tiling puzzles.

3. #### Guards in the Gallery

Place guards so that they can see into every corner of these irregular polygons.

2. 2

### Tessellations and Reptiles

M.C. Escher and beyond!

1. Included with

#### Regular Tessellations

Explore tessellation patterns that use only one type of regular polygon.

2. Included with

#### Semiregular Tessellations

What tessellations can you make when you use multiple types of shapes?

3. Included with

#### Transforming Tiles Part 1

Morph regular shapes into bizarre shapes to create new kinds of tilings.

4. Included with

#### Transforming Tiles Part 2

Extend your knowledge and skills with a second round of transformation puzzles.

3. 3

### Polyominos

Tile and cut shapes with geometrical intuition and logic.

1. Included with

#### Tiling a Chessboard

In which of these cases can you entirely cover the chessboard with dominos?

2. Included with

#### Counting All Possible Solutions

How many different ways do they fit?

3. Included with

#### Bigger Polyomino Blocks

Take a step beyond dominoes and tackle these tetrominoes and pentominoes challenges!

4. Included with

#### Challenging Packing Puzzles

Apply insight and creativity to pack these polyominos as tightly as possible.

4. 4

### Folding Puzzles

The abstract mathematics of origami.

1. Included with

#### Mathematical Origami

Unfold a paper crane and study the mountains and valleys that the folds reveal.

2. Included with

#### Dragon Folding

To make a dragon fractal, you just have keep on folding, and folding, and folding...

3. Included with

#### 1D Flat Folding

Explore the rules that govern how a single piece of paper can be folded flat.

4. Included with

#### 2D Holes and Cuts

Fold, then cut, and then unfold again to make these designs.

5. 5

### Guarding Galleries

Irregular puzzles with irregular polygons.

1. Included with

#### Strange Polygons

Get acquainted with the unusual polygons found in art gallery puzzles.

2. Included with

#### Convex vs. Concave

Study the difference between convex and concave shapes and how they affect guard placement.

3. Included with

Look specifically at cases in which the galleries are quadrilaterals and pentagons.

4. Included with

#### Efficient Guard Placement

Is there a systematic strategy for finding an ideal guard placement?

6. 6

### Pick's Theorem

Derive a wondrous theorem involving areas on a grid.

1. Included with

#### Pegboard Rectangles

Begin studying Pick's Theorem with an intuitive case.

2. Included with

#### Pegboard Triangles

What happens when you cut pegboard rectangles in half?

3. Included with