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Developments in� Pentagonal tilings

DDoC 2025 talk by Jaap Scherphuis

www.jaapsch.net

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Which polygons tile the plane?

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Which polygons tile the plane?

Every triangle tiles the plane:

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Which polygons tile the plane?

Every triangle tiles the plane:

Every quadrilateral tiles the plane:

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Which polygons tile the plane?

Every triangle tiles the plane:

Every quadrilateral tiles the plane:

Regular pentagon cannot tile:

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Which polygons tile the plane?

Every triangle tiles the plane:

Every quadrilateral tiles the plane:

Regular pentagon cannot tile:

Which pentagons do tile the plane?

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Tileable convex pentagon types

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Tileable convex pentagon types

(1918) Karl Reinhardt

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Tileable convex pentagon types

(1968) Richard B. Kershner

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Tileable convex pentagon types

(1985)

Rolf Stein

(1975) Martin Gardner

(1968) Richard B. Kershner

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Tileable convex pentagon types

(1975-77)

Richard James III

Marjorie Rice

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Tileable convex pentagon types

(1985)

Rolf Stein

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My research (2010-2013)

An interesting tiling …

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My research (2010-2013)

An interesting tiling …

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My research (2010-2013)

An interesting tiling …

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My research (2010-2013)

An interesting tiling … but it has a type 1 tile

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My research (2010-2013)

I found many tilings …

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My research (2010-2013)

I found many tilings …

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My research (2010-2013)

I found many tilings …

… all using known tile types.

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My research (2010-2013)

I found many tilings …

So no 15th type?

… all using known tile types.

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Then a 15th type was found!

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Then a 15th type was found!

(2015)�Casey Mann�Jennifer McLoud-Mann�David von Derau

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Tileable convex pentagon types

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Tileable convex pentagon types

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Tileable convex pentagon types

So is there a 16th type?

And a 17th? An 18th?

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Proof 15 convex pentagon types!

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Proof 15 convex pentagon types!

(2017) Michaël Rao proves there are only the 15 known tileable convex pentagons.

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Proof 15 convex pentagon types!

(2017) Michaël Rao proves there are only the 15 known tileable convex pentagons.

Searched for tiles rather than tilings

Split search into a finite number of cases.

Each case eliminated by computer search.

Verified by Thomas Hales.

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Proof 15 convex pentagon types!

(2017) Michaël Rao proves there are only the 15 known tileable convex pentagons.

Searched for tiles rather than tilings

Split search into a finite number of cases.

Each case eliminated by computer search.

Verified by Thomas Hales.

​

What next? Convex hexagon tiles are easy. Larger convex n-gons don’t tile. So what then?

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Next??�Tileable non-convex pentagons

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Next??�Tileable non-convex pentagons

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Next??�Tileable non-convex pentagons

I found 17 types of non-convex pentagon tiles. Are there more?

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Next??�Non-periodic tilings

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Next??�Non-periodic tilings

(2023) Dave Smith discovers an “Einstein”, a tile shape that only makes non-periodic tilings (proved by Craig Kaplan, Joseph Myers, and Chaim Goodman-Strauss)

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Next??�Non-periodic tilings

(2023) Dave Smith discovers an “Einstein”, a tile shape that only makes non-periodic tilings (proved by Craig Kaplan, Joseph Myers, and Chaim Goodman-Strauss)

Is there an Einstein with fewer sides? A pentagon?

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Thanks!

Find these tilings and more at:�jaapsch.net

Jaap Scherphuis