A Tetrad Puzzle

My recent interest in tetrads has been sparked off by an article entitled "Combines pour Tétrades," that was written by Professor Jean-Paul Delahaye, and published in the January 2020 edition of the magazine Pour La Science (N° 507). 

An interesting conversation ensued with my friend Walter Trump, who, having already done pioneer work on the subject since 1970, then decided in February 2020 to create his own specialised website "Further Questions About Tetrads." Here he gives a step-by-step analysis and relates his own tetrad story. For readers who wish to learn more about tetrads and their history, I thoroughly recommend this site, which not only gives clear explanations, but also includes an exhaustive list of references.

Definition of a Tetrad

According to Martin Gardner, in page 121 of his 1989 book "Penrose Tiles to Trapdoor Ciphers ...and the Return of Dr. Matrix," it was Michael R.W. Buckley, who, in the Journal of Recreational Mathematics (1975, Vol. 8), was the first to propose the name tetrad, for four simply connected planar regions, each pair of which sharing a finite portion of a common boundary.

Here, we will only be considering tetrads with congruent regions, each of which being transformable into each other by a combination of rigid motions that include translation, rotation, and reflection.

The following definition therefore applies: Tetrads are constituted of four congruent regions, when each of which shares a finite portion of their boundary with each of the others.

Tetrads Published in Martin Gardner's "Mathematical Games" Column

In his "Mathematical Games" column for Scientific American, Martin Gardner wrote a Problem 7.8 entitled "Exploring Tetrads," and illustrated several tetrad examples in Figure 7.11. The part D of this Figure 7.11 showed a 22-sided polygonal tetrad solution with congruent pieces that had bilateral symmetry and a bilaterally symmetric border. Martin Gardner asked if an alternative solution might exist, with polygons of fewer sides.

In the Answer 7.8 that appeared in the "Mathematical Games" column of April 1977, Martin Gardner stated that Robert Ammann, Greg Frederickson and Jean L. Loyer had each found an 18-sided polygonal tetrad with bilateral symmetry, thus improving on the 22-sided solution that he had previously published. The Ammann-Frederickson-Loyer solution, illustrated in Figure 7.33, is the smallest holeless tetrad that can made from a polyiamond with mirror symmetry. We can see that each of the four pieces of this tetrad share a common boundary with each of the others. This assembly can be achieved by the translation or the rotation of copies of a starting piece. Although reflection is also possible, no reflection of the pieces is necessary for the construction of the tetrad.

Tetrad Puzzle Transformations

While examining the different tetrads that have already been found, I noticed that some of these puzzles have completely interlocking pieces, whilst others do not. This gave me the idea of searching for jigsaw piece assemblies.

The following 5 propositions are transformations of the bilaterally symmetric tetrad found by Robert Ammann, Greg Frederickson and Jean L. Loyer. The addition of tabs, and the subtraction of blanks, can cause the jigsaw puzzle pieces to become asymmetric. Also, during the design process, we can decide whether or not certain pieces have to be flipped over, (reflected), in order to construct the tetrad:

1/ Assembly of Simple Tetrad Jigsaw Puzzle Pieces, with 3 Tabs and 3 Blanks

Tetrad Jigsaw Puzzle n°1 making a holeless tetrad: Solution n°1a.

In this first solution, which is a holeless tetrad, the pieces are not just translations and/or rotations of each other: Supposing that the starting piece is yellow, the pink and blue pieces are also reflected.

Tetrad Jigsaw Puzzle n°1 making a tetrad with a hole: Solution n°1b.

In this second solution, which is a tetrad with a hole, the pieces are always translations and/or rotations of each other.

2/ Assembly of Tetrad Jigsaw Puzzle Pieces, with 10 Tabs and 8 Blanks

Tetrad Jigsaw Puzzle n°2 making a holeless tetrad: Solution n°2a.

In this first solution, which is a holeless tetrad, the pieces are always translations and/or rotations of each other.

Tetrad Jigsaw Puzzle n°2 making a tetrad with a hole: Solution n°2b.

In this second solution, which is a tetrad with a hole, the pieces are always translations and/or rotations of each other.

3/ Assembly of Tetrad Jigsaw Puzzle Pieces, with 18 Alternate Tabs and Blanks

Tetrad Jigsaw Puzzle n°3 making a holeless tetrad: Solution n°3a.

In this first solution, which is a holeless tetrad, the pieces are not just translations and/or rotations of each other: Supposing that the starting piece is yellow, the green piece is also reflected.

Tetrad Jigsaw Puzzle n°3 making a tetrad with a hole: Solution n°3b.

In this second solution, which is a tetrad with a hole, the pieces are not just translations and/or rotations of each other: Supposing that the starting piece is yellow, the green piece is also reflected.

Observation on the limitations of tetrad puzzle pieces with tabs and blanks

When using jigsaw puzzle pieces, unless the sizes of the tabs and blanks are considerably reduced, conflicting graphics can sometimes occur in acute angles. However, it is possible to get round this difficulty by using plus (+) signs instead of tabs, and minus (-) signs instead of blanks, as shown in the examples that follow:

4/ Assembly of Tetrad Jigsaw Puzzle Pieces, with 9 Positives and 9 Negatives

Tetrad Jigsaw Puzzle n°4 making a holeless tetrad: Solution n°4a.

In this first solution, which is a holeless tetrad, the pieces are not just translations and/or rotations of each other: Supposing that the starting piece is yellow, the blue and pink pieces are also reflected.

Tetrad Jigsaw Puzzle n°4 making a tetrad with a hole: Solution n°4b.

In this second solution, which is a tetrad with a hole, the pieces are always translations and/or rotations of each other.

5/ Assembly of Tetrad Jigsaw Puzzle Pieces, with 10 Positives and 8 Negatives

Tetrad Jigsaw Puzzle n°5 making a holeless tetrad: Solution n°5a.

In this first solution, which is a holeless tetrad, the pieces are not just translations and/or rotations of each other: Supposing that the starting piece is yellow, the blue, green and pink pieces are also reflected.

Tetrad Jigsaw Puzzle n°5 making a tetrad with a hole: Solution n°5b.

In this second solution, which is a tetrad with a hole, the pieces are not just translations and/or rotations of each other: Supposing that the starting piece is yellow, the green piece is also reflected.

Observations

The necessary congruence of tetrad pieces implies that each can be transformed into each other by a combination of rigid motions which include translation, rotation, and reflection. The tetrad jigsaw puzzle exercises presented above show that, depending on the arrangement of the tabs and blanks of the pieces, reflection is sometimes, but not always needed, in order to achieve the correct assembly.

There could be a case for a new sub-classification of tetrads in general, depending on whether or not a reflection of their components is required.

In the examples given above we notice that there are jigsaw puzzle piece solutions with 9 tabs (+) and 9 blanks (-), with 10 tabs (+) and 8 blanks (-), or with 8 tabs (+) and 10 blanks (-). Why do these differences occur? The following document is an analysis of the different combinations of the positives (+) and negatives (-) of the tetrad puzzle pieces, depending the choices made during the design process:



We can also construct other shapes that are not necessarily tetrads. Readers who are interested, are invited to explore the many possibilities, and create their own patterns.

Acknowledgements

I wish to thank Craig Knecht and Walter Trump for their encouragements and useful comments during the preparatory phase of this post.