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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.

Interactive
quizzes

36

Concepts and
exercises

280+
  1. 1

    Introduction

    Sample the beautiful side of geometry.

    1. Polyomino Tiling

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

    2. Infinite Areas

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

    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
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      Regular Tessellations

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

    2. Included with
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      Semiregular Tessellations

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

    3. Included with
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      Transforming Tiles Part 1

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

    4. Included with
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      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
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      Tiling a Chessboard

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

    2. Included with
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      Counting All Possible Solutions

      How many different ways do they fit?

    3. Included with
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      Bigger Polyomino Blocks

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

    4. Included with
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      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
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      Mathematical Origami

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

    2. Included with
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      Dragon Folding

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

    3. Included with
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      1D Flat Folding

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

    4. Included with
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      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
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      Strange Polygons

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

    2. Included with
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      Convex vs. Concave

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

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      Quadrilateral and Pentagonal Galleries

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

    4. Included with
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      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
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      Pegboard Rectangles

      Begin studying Pick's Theorem with an intuitive case.

    2. Included with
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      Pegboard Triangles

      What happens when you cut pegboard rectangles in half?

    3. Included with
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      Pick's Theorem Generalized

      Prove Pick's Theorem for any lattice polygon.

    4. Included with
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      Pick's Theorem With One Hole

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