Dani Laura (they/she/he)<p>I have found a novel family of rep-tiles which produce aperiodic tilings. The prototile is a triangle with smallest side 1 and biggest side 2, the other side is 1 < x <= 2. The family includes one pointed isosceles triangle, the right triangle of angles 30-60-90 (half an equilateral triangle), and other scalene, obtuse or acute, triangles. The first image shows relevant members of the family, the second the substitution rule. The isosceles triangle of the family has another already known aperiodic tiling ( <a href="https://tilings.math.uni-bielefeld.de/substitution/viper/" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">tilings.math.uni-bielefeld.de/</span><span class="invisible">substitution/viper/</span></a> ) which looks the same but is different because there the tile has no reflections, whereas here some tiles are reflected (in the case of the isosceles triangle the reflection makes a difference when applying the substitution). Figure 3 shows the difference between that tessellation and the one proposed here, mine has just four slopes. Last figure shows a zoom into one big instance of the tiling for the right triangle.<br><a href="https://mathstodon.xyz/tags/TilingTuesday" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>TilingTuesday</span></a> <a href="https://mathstodon.xyz/tags/tiling" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>tiling</span></a> <a href="https://mathstodon.xyz/tags/Mathart" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Mathart</span></a> <a href="https://mathstodon.xyz/tags/geometry" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>geometry</span></a> <a href="https://mathstodon.xyz/tags/Mathematics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Mathematics</span></a></p>