Saturday, May 5, 2018

Tessellating activities

It has been over a year since I last published a book via CreateSpace, and since then CreateSpace has ceased to sell books; all sales are now through Amazon. If you click on any CreateSpace link in a previous post, it will take you to an Amazon page.

I have not spent much time or effort with tessellations (or mazes) since Tessellating Animals Activity Book was published in January 2017. A post at the time tried to indicate what kind of activities were in the book, but the illustrations were of partial puzzles that could not be solved. The sample pages that Amazon shows are even less revealing of what is in the book.

Below are examples of some of the types of puzzles in Tessellating Animals Activity Book. First is a mini-Sudoku. Normal Sudoku puzzles are 9-by-9 grids, but simpler versions can be constructed with 4-by-4 or 6-by-6 grids. The puzzle below uses numbers 1 through 6. All rows and all columns must have each number only once. In addition, there are six boxes indicated by the shading and each of those boxes must also have each of the six numbers only once.

The book contains 14 mini-Sudoku puzzles. (The bird-head shape was in preliminary versions of the book but was dropped before it was published because it is a low-quality design.)

A second type of puzzle is a word search, which also use grids. The grid here is provided by a weird shape that was never in any version of the book but which works well for this type of puzzle. Words can be vertical, horizontal, or diagonal and they may be reversed. Find the following words: reflect, valence, hexagon, rotate, escher, glide, flips, edge. The letters not used in any of these words will spell another word that is related to tessellations.

There are ten word-search puzzles in the book and all are larger than this one.

There are eleven pages of decoder puzzles in the book and most of those pages have more than one puzzle. In the puzzle below the letters have been replaced by their position in the alphabet: A=1, B=2, C=3, etc. 
This shape was a first attempt to find a standing bird that fit Heesch type TGTG. It is not very good and I have found better ways to illustrate the type.

The book contains a number of mazes, some of the traditional variety but others that are coded. They are simpler than the coded mazes in the book Hidden Path Mazes: Decode to Solve.

The various puzzles and activities are supplemented with explanations of basic characteristics of tessellations.

Friday, April 20, 2018

Tessellating wine glasses

Goblets, wine glasses, or chalices are easy to tessellate and if you search the Internet for images you should be able to find at least two ways in which the shape fits into a tiling pattern. Below is one of these ways.
The five-edged goblets are an example of isohedral class IH26. IH26 is the result of bisecting isohedral class IH17. If the straight bottom is considered an edge of central rotation, the pattern fits Heesch type TCTCC. If it is an edge of reflection, it is not a Heesch type.

If we slide alternate rows, each glass has six neighbors rather than the five in the pattern above.
When the six-edged goblets are centered over each other as in the image above, we have an example of isohedral class IH15, which simultaneously fits Heesch types TCCTCC, TCCTGG, and TG1G1TG2G2.

The other easy way to tessellate this shape is to stack them in the pattern below.

This arrangement is an example of isohedral class IH12 that simultaneously fits Heesch types TTTTTT and TG1G1TG2G2. Four of the edges are identically shaped.

All six edges can be identically shaped and the image still resembles a chalice or wine glass.

However, this shape can also be fit together in a different way, one that I have not found on the Internet.

It is an example of Heesch type TG1G1TG2G2 but not of TTTTTT. Can you find the TT pair?