# 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.
Michael R. W. Buckley first used the name *tetrad* in
an article in the *Journal of Recreational Mathematics*, volume 8.
Martin Gardner's book *Penrose Tiles to Trapdoor Ciphers*
(Freeman, 1989; ISBN 0-7167-1987-8) 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 Tetrads.
The smallest polyhop or polybrick tetrad uses 7-hops:

### Holeless

The smallest polyhops that form holeless
tetrads use 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.

The smallest polydrafter tetrad,
allowing cells off the polyiamond grid, uses a tetradrafter:

The smallest polydrafter with all cells on the polyiamond grid
that forms a tetrad with some cells off the grid is this hexadrafter:

The smallest symmetric polydrafter that forms a tetrad is this
extended hexadrafter:

The smallest polydrafters on the polyiamond grid that form tetrads
on the grid are these octadrafters:

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:

Livio Zucca found
the fewest edges
for a polygon that can form a tetrad:

*Last revised 2018-11-12.*

Back to Polyform Curiosities.

Col. George Sicherman
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