Polyomino Tiling
Get your first taste of beautiful geometry by exploring these tiling puzzles.
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.
Sample the beautiful side of geometry.
M.C. Escher and beyond!
Explore tessellation patterns that use only one type of regular polygon.
What tessellations can you make when you use multiple types of shapes?
Morph regular shapes into bizarre shapes to create new kinds of tilings.
Extend your knowledge and skills with a second round of transformation puzzles.
Some very strange shapes can tile the plane, but the tessellations they make can be truly bizarre.
Make larger copies from smaller copies.
Learn how geometric series can be summed up geometrically!
Tile and cut shapes with geometrical intuition and logic.
In which of these cases can you entirely cover the chessboard with dominos?
How many different ways do they fit?
Take a step beyond dominoes and tackle these tetrominoes and pentominoes challenges!
Apply insight and creativity to pack these polyominos as tightly as possible.
What if you can only use the X-shaped pentomino?
Investigate methods for finding out if a tiling is possible.
Cut shapes into several pieces that are identical.
Test your mettle with this polyomino finale.
The abstract mathematics of origami.
Unfold a paper crane and study the mountains and valleys that the folds reveal.
To make a dragon fractal, you just have keep on folding, and folding, and folding...
Explore the rules that govern how a single piece of paper can be folded flat.
Fold, then cut, and then unfold again to make these designs.
Mathematically, how can you tell if something is flat foldable?
Extend your exploration of flat folding one final step further.
Irregular puzzles with irregular polygons.
Get acquainted with the unusual polygons found in art gallery puzzles.
Study the difference between convex and concave shapes and how they affect guard placement.
Look specifically at cases in which the galleries are quadrilaterals and pentagons.
Is there a systematic strategy for finding an ideal guard placement?
Practice making galleries that are tough to guard.
Fisk's proof is puts an upper bound how many guards you might need for an n-sided gallery.
Investigate internal walls and other twists that you can add to the art gallery puzzle.
Derive a wondrous theorem involving areas on a grid.
Begin studying Pick's Theorem with an intuitive case.
What happens when you cut pegboard rectangles in half?
Prove Pick's Theorem for any lattice polygon.
Poke a hole in your polygons and see what formula comes out.
Extend Pick's Theorem one more time to address this multi-holed variation.