# Beautiful Geometry

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

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#### Overview

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.

### Topics covered

• Convexity and Concavity
• Deconstructing Origami
• Fisk's Coloring Proof
• Fractals
• Lattice Polygons
• Packing Puzzles
• Pick's Theorem
• Polyominoes
• Reptiles
• Tessellations
• The Art Gallery Problem
• Triangulation

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1. 1

## Introduction

Sample the beautiful side of geometry.

1. ## Polyomino Tiling

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

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2. ## Infinite Areas

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

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3. ## Guards in the Gallery

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

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2. 2

## Tessellations and Reptiles

M.C. Escher and beyond!

1. ## Regular Tessellations

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

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2. ## Semiregular Tessellations

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

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3. ## Transforming Tiles Part 1

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

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4. ## Transforming Tiles Part 2

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

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5. ## Irregular Tiles

Some very strange shapes can tile the plane, but the tessellations they make can be truly bizarre.

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6. ## Reptiles

Make larger copies from smaller copies.

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7. ## Infinite Arithmetic

Learn how geometric series can be summed up geometrically!

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3. 3

## Polyominos

Tile and cut shapes with geometrical intuition and logic.

1. ## Tiling a Chessboard

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

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2. ## Counting All Possible Solutions

How many different ways do they fit?

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3. ## Bigger Polyomino Blocks

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

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4. ## Challenging Packing Puzzles

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

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5. ## X-Only

What if you can only use the X-shaped pentomino?

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6. ## Tiling and Cutting

Investigate methods for finding out if a tiling is possible.

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7. ## Congruent Cutting

Cut shapes into several pieces that are identical.

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8. ## Battle of the Four Oaks

Test your mettle with this polyomino finale.

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4. 4

## Folding Puzzles

The abstract mathematics of origami.

1. ## Mathematical Origami

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

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2. ## Dragon Folding

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

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3. ## 1D Flat Folding

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

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4. ## 2D Holes and Cuts

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

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5. ## 2D Single-Vertex Flat Folding (I)

Mathematically, how can you tell if something is flat foldable?

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6. ## 2D Single-Vertex Flat Folding (II)

Extend your exploration of flat folding one final step further.

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5. 5

## Guarding Galleries

Irregular puzzles with irregular polygons.

1. ## Strange Polygons

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

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2. ## Convex vs. Concave

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

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3. ## Quadrilateral and Pentagonal Galleries

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

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4. ## Efficient Guard Placement

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

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5. ## Worst-Case Designs

Practice making galleries that are tough to guard.

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6. ## Fisk's Coloring Proof

Fisk's proof is puts an upper bound how many guards you might need for an n-sided gallery.

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7. ## Further Art Gallery Research

Investigate internal walls and other twists that you can add to the art gallery puzzle.

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6. 6

## Pick's Theorem

Derive a wondrous theorem involving areas on a grid.

1. ## Pegboard Rectangles

Begin studying Pick's Theorem with an intuitive case.

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2. ## Pegboard Triangles

What happens when you cut pegboard rectangles in half?

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3. ## Pick's Theorem Generalized

Prove Pick's Theorem for any lattice polygon.

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4. ## Pick's Theorem With One Hole

Poke a hole in your polygons and see what formula comes out.

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5. ## Pick's Theorem With Multiple Holes

Extend Pick's Theorem one more time to address this multi-holed variation.

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