Fractal self-replicating tile formed by iterating a 3-kite

A fractal self-replicating tile created by iterating a pentabolo

Pavers in Negombo, Sri Lanka, photogaraphed by a friend earlier today

Another √6-scaled fractal self-replicating tile. This one is based on a rhombic grid in which the grid cells have diagonals measuring 1 and √15/3.

Using square- or triangle-based grids, there are no overlapping transformations of the grid for n = 6. By distorting a grid, for example stretching a square grid by √2, n=6 becomes allowed. Here a hexa-rectangle is used to form a fractal self-replicating tile.

Charles O Perry sculpture in the lobby of the Hyatt Regency in San Francisco

I'm almost certain the answer is yes.

Fractal self-replicating tile created by iterating a tetrakite seven times

Wth Sujan Shrestha at the Computer Museum, in Hunt Valley, MD last week. This is one of around 70 Apple-1 computers whose whereabouts are known, out of around 200 total made.

The online gallery for 2026 Joint Mathematics Meetings Exhibition of Mathematical Art is now live: gallery.bridgesmathart.org/exhibitions/....

Some pavers spotted in Cairo last month

A Truchet cafe, in Cairo last month

Ancient Egyptian ceiling with a star motif

5-stars were commonly used on the ceilings of tombs and other buildings in Ancient Egypt to depict the heavens. They are usually not connected to one another, but they are connected in this ceiling in the Ptolemaic Denderah Temple to form a sort of tessellation of hexagons and quadrilaterals.

Ancient Egyptian numbers used to track sacrifices on the side of the Temple of Ramesses III (1217 BC - 1155 BC) in Luxor. A vertical tick mark denotes 1 and an inverted U ten.

Great "tessellation" on the ceiling of the 20th Dynasty (ca 1100 BC) Tomb of Anherkha in the Valley of the Workers in Luxor

Another type of Cairo pentagon paver in Cairo.

Everyone is invited to attend the SCULPT 2025: Show & Tell, a digital gathering that celebrates creative exploration across art, geometry, design, and fabrication.
Date: Friday, November 7, 2025
Time: 8:00 – 10:30 AM (PST)
Zoom Link: cca.zoom.us/j/9826107944...

Polyaboloes, polygons formed by joining isosceles right triangles in edge-to-edge fashion, can also be arranged iteratively to form fractal self-replicating tiles. A particular tetrabolo was used to form these two reptiles, with mirroring between successive iterations employed in the right one.

Two fractal self-replicating tiles formed by iterating a 4-rhomb. On the right with mirroring between successive iterations.

This weekend would be a great time to submit your art for the exhibition to be held at the Joint Mathematics Meetings. Take advantage of this opportunity to share your work with over 5000 attendees this January in Washington, DC. You’ll need to submit by the 15th at gallery.bridgesmathart.org.

Polyrhombs, formed by joining 60°-120° rhombi in edge-to-edge fashion, can also be iterated to form self-replicating tiles. A 3-rhomb iterates to the same fractal as the envelope of the terdragon curve. Mirroring between successive iterations yields a very different looking reptile (right).

I just finished a new website on fractal self-replicating tiles ("reptiles") based on polyominoes, polyhexes, and polyiamonds: www.mathartfun.com/fractaldiver...

New sculpture illustrating the fact that cutting a three-half-twists band lengthwise results in a trefoil knot. Unglazed ceramic and leather cord.

Submissions are now being accepted for the Exhibition of Mathematical Art to be held as part of the Joint Mathematics Meetings, Washington, DC, in January of 2026. Apply online through October 15 at gallery.bridgesmathart.org. pic.x.com/MGSEyexSzo

Alpha Omega d24 YouTube video by Henry Segerman

youtu.be/fgMoxEAZtoA?...
New from The Dice Lab; available at mathartfun.com/DiceLabDice....

A new article is out of which I'm a coauthor: "Visualization of Escher-like hyperbolic tessellations": www.sciencedirect.com/science/arti...

To submit your work, please fill out the Google Form at docs.google.com/forms/d/1SQA...

Submissions are now being accepted through September 14 for SCULPT 2025. Accepted work will be presented in a ZOOM Show and Tell event on November 7. In addition, works will be digitally showcased in the exhibition section of SCULPT 2025 and in a proceedings volume.

This mixed-media piece consists of an unglazed black-clay ceramic band with two half twists, making it a two-sided object. Steel dowels with long and short ends are spaced along the centerline to define a second band with two half twists, orthogonal to the first.

Windows, Uzekistan