This model consists of three “meta-tiles” constructed from einstein hat tiles. The four are described below.  

Three fundamental results relating to Einstein tiles are:

1. They do indeed tile the plane. 
2. They tile aperiodically, and
3. They do not tile periodically.

A proof of the first result involves a recursive construction of “supertiles.” 

The basis of the recursion is a collection of four “meta-tiles” that are constructed with individual “hats.”  They are

• The H-block, made up of one inverted hat and three non-inverted hats. The only inverted hats used in the construction are here. 
• The T-block, which is a single non-inverted hat.
• The P-block, which is made of a pair of non-inverted hats.
• The F-block, which is identical to the P-block at the first level. At higher levels, the P and F blocks are distinct. 

You’ll probably want to print P and F blocks with different colors. 

On inversion:  The hat has no symmetries. It seems to be standard to view the basic hat as having its higher peak to the right. It’s mirror image, which you get by flipping it over, has the higher peak on the left. 

The image with multiple blocks is the second level H-block. To construct the third  level H-block, you need to print 22 first level H-blocks, 3 first level T-blocks, 15 first level P-blocks, and 24 first level F-blocks. 

In addition to the tiles that can be printed in multiple copies to construct higher level blocks, a model of the hat, which is a union of kites in the shape of a sixth of a regular hexagon, is included. This hat is scaled somewhat larger and can’t be used in the build of supertiles unless you rescale it. 

All models were created with Mathematica.

IMPORTANT:  If using .stl files, scale down to 78.74% and then scale z-axis by 140% to be consistent with othe files.