The Hexayurt is an incredible project, reinventing disaster housing:
hexayurt.com

Vinay Gupta, the creator of the hexayurt approached me to try to find larger shapes that could also be build (reasonably) simply with zero-waste from 2x1 rectangles. His version of the story is here: vinay.howtolivewiki.com/blog/other/large-hexayurt-style-domes-a-problem-solved-1730

I came up with two additional larger domes. Models for the original hexayurt and the two new domes are all here.

A longer write up and a little more detail are here:
tilings.org.uk/Hexayurt_Family.pdf

A single tile that can fill three dimensional space but never gives a periodic tiling. For more details see:

blog.makezine.com/archive/2010/03/worlds_first_aperiodic_tiling_with.html

maxwelldemon.com/2010/04/01/socolar_taylor_aperiodic_tile/

A blender file with the python code constructing the model is here:
tilings.org.uk/monotile_2.blend

Can this shape tile the plane? It is not easy to guess by just putting the shapes together. With a little thought however there is an easy proof. It can't. How far can it go though? How many rings of tiles can you build around a first one. This is called the Heesch number of the tiling.

This shape is a world record holder you need 20 tiles before it will tile the plane periodically (10 up to symmetry). It fits together in an amazing variety of ways but can easily block itself. Making it a fun puzzle.

The tiles were found by a massive computer search by Joseph Myers. Look for this and other puzzling shapes at: srcf.ucam.org/~jsm28/tiling/

The idea to make physical tiles was originally from Chaim Goodman-Strauss: mathbun.com