Math News and Discussions

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AI finds 3D shape that fills all space with a never-repeating pattern

A 3D shape that can tile 3D space, leaving no gaps and never forming a repeating pattern, has been discovered using an AI model. The researcher behind the findings, who has no academic background in the area, says his work has angered professional mathematicians

By Matthew Sparkes
28 September 2026

In 2023, after decades of searching, mathematicians discovered a shape called “the hat” – so named for its passing resemblance to a fedora – that can tile a 2D surface without gaps and without ever creating a repeating pattern. Now, a researcher has used AI to go one dimension better and discover a 3D shape that can do the same thing.

Simple shapes like squares can tile a 2D surface, but will form repeating patterns. So-called aperiodic tiles achieve the former without the latter – you can scroll in any direction, infinitely, and not find a regularly repeating pattern.

That sounds like a nifty but ultimately useless mathematical curiosity, but since the 2023 publication of the 2D tile, it has been used in scientific papers in fields from engineering to chemistry. Some have explored the likely physical properties of a material with hat-shaped crystals. Others have found that structures built using hat-shaped building blocks could be more resistant to fracturing than those built using famously strong honeycomb-like building blocks.

Now, Ioannis Tsiokos, a software developer with no background in mathematics, has used AI to discover a 3D monotile he calls Chair44. He did so by describing the 2D problem to OpenAI’s GPT-6 Astra model and asking it to find a similar result in three dimensions.

“Astra found this, I didn’t find the tile,” says Tsiokos. “I gave Astra the theory, I gave it the laws, I gave it the theorems, the Lean code, the papers, and I said: ‘Here’s the problem.’ It’s kind of cheeky: I’ve been in forums with other mathematicians, and [of] course they’re all angry with me… because I don’t have the background to land this result.”

https://www.newscientist.com/article/25 ... g-pattern/



The Chair44 3D monotile. Credit: Ioannis Tsiokos

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Gatekeepers who gatekeep any field just to feel special are the scum of the earth

Imagine holding back human advancement just because some bloke burst your bubble

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Myth or Math? Is the Nautilus Shell’s Golden Ratio Real?
By Cybele Tamulonis,
September 24, 2026

Introduction:

(Science Friday) The chambered nautilus has inspired plenty of big clams, we mean, claims! Its coiled shell is commonly pictured alongside the golden ratio, a number associated with spirals, rectangles, art, architecture, and patterns in nature. But placing two similar-looking shapes next to each other isn’t enough to prove a mathematical relationship.

Just as scientists in the video below tested assumptions about whether nautiluses can learn, you can test this shell claim using evidence. In this math activity, you’ll measure images of real nautilus shells, calculate their proportions, and determine whether they really are “golden,” or whether this familiar idea is more myth than math!

Additional Extract:

The shell resembles a logarithmic spiral, a curve that expands while maintaining a similar overall shape. But not every logarithmic spiral is a golden spiral.

A golden spiral is based on the golden ratio, usually represented by the Greek letter phi, or 𝜙.
𝜙 = 1.618

…
Because a nautilus shell expands as it grows, people frequently claim that it forms a perfect golden spiral. However, Christopher Bartlett, an art professor from Towson University, decided to test that assumption. He found that measurements of 80 nautilus shells in the Smithsonian National Museum of Natural History collection showed an average length-to-height ratio of approximately 1.31, not 1.618. That’s a 19% difference! Usually, a 5-10% difference is considered acceptable for experimental results, so his findings cast doubt on the assumption that the nautilus shell follows the golden ratio.

Read more here: https://www.sciencefriday.com/educatio ... -ratio /

The golden ratio is closely connected with the Fibonacci sequence—1, 1, 2, 3, 5, 8, 13, …—where each number is the sum of the two before it. As the sequence grows, the ratio of one number to the previous one gets closer and closer to ϕ. Thus ϕ is the limit of these ratios.

Source: https://www.britannica.com/science/golden-ratio

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