# Polyform Tetrads

Then clap four slices of pilaster on 't,

That laced with bits of rustic makes a front.

—Alexander Pope,
Epistle to Richard Boyle

A *tetrad* is a plane figure made of four congruent shapes,
joined so that each shares a boundary with each.
The first to study tetrads was Walter Trump, in 1970 or 1971.
He recorded his findings in notebooks, but did not publish them.
In 2020 he presented them on his website
Further
Questions About Tetrads.
It contains a great deal of general and
specific information about tetrads.

Michael R. W. Buckley first used the name *tetrad* in
a 1975 article in the *Journal of Recreational Mathematics*, volume 8.
In 1979 Trump sent many of his results to Martin Gardner,
including the holeless 11-omino tetrad shown under Polyominoes and Polynars.

Martin Gardner's book *Penrose Tiles to Trapdoor Ciphers*
(Freeman, 1989; ISBN 0-7167-1987-8) also shows
some plane constructions by Scott Kim,
including a holeless tetrad for a 12-omino,
a holeless tetrad for a 26-iamond
with mirror symmetry, and a holeless tetrad for a tetrahex.
Karl Scherer shows many varieties of tetrads at
Wolfram.

Here I consider only tetrads that are themselves polyforms of the same
type as their tiles.

See Polyomino and Polynar Tetrads.
The smallest polyhop or polybrick tetrad uses 7-hops:

### Holeless

The smallest polyhops that form holeless
tetrads are 10-hops:

For *similar* or *scaled tetrads,*
in which the four pieces are similar but need not be congruent,
see Scaled Polytan Tetrads.
The smallest polyabolo tetrads, found by Juris
Čerņenoks, use 12-aboloes:

### Holeless

Juris Čerņenoks found
the smallest holeless polyabolo tetrads, using 16-aboloes:

### Symmetric Tiles

The smallest tetrad for a symmetric polyabolo
uses 22-aboloes:

The smallest tetrad for a polyabolo with mirror symmetry uses a 26-abolo
or 13-omino:

Dr. Karl Scherer found
this holeless tetrad for a symmetric polyabolo.
It uses a 278-abolo with only 27 edges:

See Polyiamond Tetrads.
The two smallest polypent tetrads use a hexapent:

### Symmetric Tiles

The smallest tetrads with symmetric polypents use 11-pents:

The smallest tetrad for a polypent with bilateral symmetry about an edge
uses a 12-pent:

The smallest tetrad for a polypent with birotary symmetry
uses a 12-pent:

See Polyhex Tetrads.
The smallest polyhept tetrad uses 9-hepts:

The smallest tetrad for a symmetric polyhept uses 15-hepts:

The smallest tetrads for polyhepts with mirror symmetry around an edge
use 16-hepts:

The smallest polyoct tetrads use 6-octs:

The smallest tetrads for a symmetric polyoct use 10-octs:

The smallest tetrads for polyocts with diagonal symmetry use 13-octs:

Juris Čerņenoks found the smallest polykite tetrad, using
7-kites …

### Symmetric Tiles

… and the smallest tetrad with symmetric polykites, using 8-kites:

The smallest tetrads with symmetric polykites of odd order
use 13-kites:

### Holeless

Juris Čerņenoks found the smallest holeless polykite
tetrad, using 9-kites:

The smallest holeless tetrad with symmetric polykites uses 16-kites:

The smallest polycairo tetrads use a 7-cairo:

### Holeless

The smallest holeless polycairo tetrad uses a 9-cairo:

### Symmetric Tiles

The smallest symmetric polycairo that forms a tetrad
is this 16-cairo:

The smallest tetrad for polycairos with bilateral symmetry
uses 18-cairos.

See Polydrafter Tetrads.
For *similar* or *scaled tetrads,*
in which the four pieces are similar but need not be congruent,
see Scaled Polydom Tetrads.
The smallest polydom tetrad uses a hexadom:

There are 28 heptadoms that can form tetrads:

Around 1979 Walter Trump found the the fewest edges
for a polygon that can form a tetrad:

Later Livio Zucca found another pentagon solution:

*Last revised 2020-03-18.*

Back to Polyform Curiosities.

Col. George Sicherman
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