# orderedItems[].object.content
See also the resources page for this paper: https://cs.uwaterloo.ca/~csk/spectre/

# orderedItems[].object.contentMap.en
See also the resources page for this paper: https://cs.uwaterloo.ca/~csk/spectre/

# orderedItems[].object.content
In plane tiling, it is completely standard that tiles may be reflected; nevertheless, some people were dissatisfied that the aperiodic hat monotile requires reflections to tile the plane. In our new preprint, we present the Spectre, the first example of a vampire einstein: an aperiodic monotile that tiles the plane without reflections. https://arxiv.org/abs/2305.17743

# orderedItems[].object.contentMap.en
In plane tiling, it is completely standard that tiles may be reflected; nevertheless, some people were dissatisfied that the aperiodic hat monotile requires reflections to tile the plane. In our new preprint, we present the Spectre, the first example of a vampire einstein: an aperiodic monotile that tiles the plane without reflections. https://arxiv.org/abs/2305.17743

# orderedItems[].object.content
This article on the aperiodic monotile includes a quote from Robert Berger who found the very first aperiodic set of tiles (thanks to @11011110 for tracking down his subsequent career in a 2011 Wikipedia deletion discussion): https://www. nytimes.com/2023/03/28/science/mathematics-tiling-einstein.html

# orderedItems[].object.contentMap.en
This article on the aperiodic monotile includes a quote from Robert Berger who found the very first aperiodic set of tiles (thanks to @11011110 for tracking down his subsequent career in a 2011 Wikipedia deletion discussion): https://www. nytimes.com/2023/03/28/science/mathematics-tiling-einstein.html

# orderedItems[].object.content
@Danpiker @nilesjohnson @christianp My conclusion was that for a two-dimensional topological disk monotile, the only flexibility is varying the side lengths as discussed in the paper; you can't change the sides away from straight lines or change the angles. The metatile shapes are much more flexible, however.

# orderedItems[].object.contentMap.en
@Danpiker @nilesjohnson @christianp My conclusion was that for a two-dimensional topological disk monotile, the only flexibility is varying the side lengths as discussed in the paper; you can't change the sides away from straight lines or change the angles. The metatile shapes are much more flexible, however.

# orderedItems[].object.content
@robinhouston That agrees with the figure Adam came up with (computed as a ratio of unreflected to reflected rather than as a proportion of all tiles): https://cp4space.hatsya.com/2023/03/21/aperiodic-monotile/

# orderedItems[].object.contentMap.en
@robinhouston That agrees with the figure Adam came up with (computed as a ratio of unreflected to reflected rather than as a proportion of all tiles): https://cp4space.hatsya.com/2023/03/21/aperiodic-monotile/

# orderedItems[].object.content
@barrygoldman1 @csk @tao Figure 3.2.4(a) in Tilings and Patterns is a periodic tiling by the Voderberg tile.

# orderedItems[].object.contentMap.en
@barrygoldman1 @csk @tao Figure 3.2.4(a) in Tilings and Patterns is a periodic tiling by the Voderberg tile.

# orderedItems[].object.content
@robinhouston Open as well, and probably much harder to resolve than the case with only 180-degree rotations.

# orderedItems[].object.contentMap.en
@robinhouston Open as well, and probably much harder to resolve than the case with only 180-degree rotations.

# orderedItems[].object.content
@robinhouston As far as I know it's open whether there could be an aperiodic monotile that tiles using only translates and copies under 180-degree rotation (Schattschneider asked whether all tiles that tile like that must satisfy the Conway criterion, c.f. the results of Girault-Beauquier and Nivat and Kenyon for tilings using translates only).

# orderedItems[].object.contentMap.en
@robinhouston As far as I know it's open whether there could be an aperiodic monotile that tiles using only translates and copies under 180-degree rotation (Schattschneider asked whether all tiles that tile like that must satisfy the Conway criterion, c.f. the results of Girault-Beauquier and Nivat and Kenyon for tilings using translates only).

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@markuswerle @csk My diagrams were done in TikZ (sources included in the paper sources on the arXiv), the other authors used other tools.

# orderedItems[].object.contentMap.de
@markuswerle @csk My diagrams were done in TikZ (sources included in the paper sources on the arXiv), the other authors used other tools.

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@benjohn@noparticular.place @csk The idea for those proof structure diagrams came from e.g. Greenfeld-Tao 2022 (see page 5 of their preprint). https://arxiv.org/abs/2211.15847

# orderedItems[].object.contentMap.en
@benjohn@noparticular.place @csk The idea for those proof structure diagrams came from e.g. Greenfeld-Tao 2022 (see page 5 of their preprint). https://arxiv.org/abs/2211.15847

# orderedItems[].object.content
An aperiodic monotile (joint work with David Smith, @csk and Chaim Goodman-Strauss): https://arxiv.org/abs/2303.10798

# orderedItems[].object.contentMap.en
An aperiodic monotile (joint work with David Smith, @csk and Chaim Goodman-Strauss): https://arxiv.org/abs/2303.10798