Interactive course — Foundational Math

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

Interactive quizzes

36

Concepts and exercises

280+

Course map

Prerequisites and Next Steps

  1. 1

    Introduction

    Sample the beautiful side of geometry.

    1. Polyomino Tiling

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

      1
    2. Infinite Areas

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

      2
    3. Guards in the Gallery

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

      3
  2. 2

    Tessellations and Reptiles

    M.C. Escher and beyond!

    1. Regular Tessellations

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

      4
    2. Semiregular Tessellations

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

      5
    3. Transforming Tiles Part 1

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

      6
    4. Transforming Tiles Part 2

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

      7
    5. Irregular Tiles

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

      8
    6. Reptiles

      Make larger copies from smaller copies.

      9
    7. Infinite Arithmetic

      Learn how geometric series can be summed up geometrically!

      10
  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?

      11
    2. Counting All Possible Solutions

      How many different ways do they fit?

      12
    3. Bigger Polyomino Blocks

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

      13
    4. Challenging Packing Puzzles

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

      14
    5. X-Only

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

      15
    6. Tiling and Cutting

      Investigate methods for finding out if a tiling is possible.

      16
    7. Congruent Cutting

      Cut shapes into several pieces that are identical.

      17
    8. Battle of the Four Oaks

      Test your mettle with this polyomino finale.

      18
  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.

      19
    2. Dragon Folding

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

      20
    3. 1D Flat Folding

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

      21
    4. 2D Holes and Cuts

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

      22
    5. 2D Single-Vertex Flat Folding (I)

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

      23
    6. 2D Single-Vertex Flat Folding (II)

      Extend your exploration of flat folding one final step further.

      24
  5. 5

    Guarding Galleries

    Irregular puzzles with irregular polygons.

    1. Strange Polygons

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

      25
    2. Convex vs. Concave

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

      26
    3. Quadrilateral and Pentagonal Galleries

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

      27
    4. Efficient Guard Placement

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

      28
    5. Worst-Case Designs

      Practice making galleries that are tough to guard.

      29
    6. Fisk's Coloring Proof

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

      30
    7. Further Art Gallery Research

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

      31
  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.

      32
    2. Pegboard Triangles

      What happens when you cut pegboard rectangles in half?

      33
    3. Pick's Theorem Generalized

      Prove Pick's Theorem for any lattice polygon.

      34
    4. Pick's Theorem With One Hole

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

      35
    5. Pick's Theorem With Multiple Holes

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

      36