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Last Notes

2026-11-05 16:23:33 GMT

π’s Palindromic Prime
This 53-digit number (31415926535897932384626433833462648323979853562951413) is a palindromic prime formed by taking digits of π and reflecting them. It was discovered by G. L. Honaker, Jr. in 1999 (tracked as OEIS sequence A039954).
☞ More number facts from our website.
https://archimedes-lab.org/2026/11/04/%cf%80s-palindromic-prime/
https://archimedes-lab.org/wp-content/uploads/2026/11/Palindromic-pi-prime.png
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2026-10-26 15:42:58 GMT

A Subtle Visual Trick: The Hering Illusion
When two straight, parallel lines are placed against a radial background, like the spokes of a bicycle wheel, they appear to bow outward. This classic geometrical illusion was described by German physiologist Ewald Hering in 1861.
I often use this effect in my work, including book illustrations, because it lends itself to all kinds of creative variations.
Take the candles shown below: they appear to bow outward at the center, yet their outer edges are perfectly straight and vertical. The secret lies in the radial rays spreading from the central light, which distort our perception of the parallel lines.
Below is a variation I created in the 1990s using China ink and watercolor.
In another version, I applied the same principle to an aerial view of buildings. The parallel lines appear slightly bowed, even though they remain perfectly straight. A striking example of how context can change the way we see geometry.
These variations demonstrate how a simple geometric illusion can inspire entirely different visual experiences, revealing just how easily our eyes can be deceived.
https://archimedes-lab.org/2026/10/25/a-subtle-visual-trick-the-hering-illusion/
https://archimedes-lab.org/wp-content/uploads/2026/10/Hering_illusion-188x300.png
https://archimedes-lab.org/wp-content/uploads/2026/10/Candles_illusion-1024x737.jpg
https://archimedes-lab.org/wp-content/uploads/2026/10/Candles_illusion2_Drawing-1024x961.jpg
#ewaldhering #illusion #below #german #china
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2026-10-17 03:04:03 GMT

How AI Mirrors Our Biases
Humans are far less random than they think. When asked to pick a number from 1 to 10, we tend to favor 7, sometimes choose 5, and avoid the extremes.
LLMs (Large Language Models) learn from human-generated text, so their responses can reflect our biases. Rather than choosing randomly, they predict likely answers based on patterns learned during training. A process called gradient descent gradually adjusts the model’s parameters to improve its predictions, potentially reproducing or amplifying our preferences.
In a truly random process, every number has an equal probability of being selected:
P(number) = 1/10 = 10%
Human choices tell a different story. In some surveys, about 33% of people pick 7—more than three times the 10% expected by chance. Why 7? Superstition, cultural associations, and psychological preferences may all play a role. People also tend to favor odd numbers, especially primes. Even dice games reinforce its familiarity: with two standard six-sided dice, 7 is the most likely sum, with a probability of 6/36, or about 16.7%.
The graph illustrates the difference between uniform randomness, human choices, and AI predictions. An LLM doesn’t inherently generate truly random numbers; its answers depend on learned patterns, the prompt, and its settings. If its training data reflects our preference for 7, that preference may influence its response, too.
In short, AI can reproduce our biases while making them look like statistical certainty.
https://archimedes-lab.org/2026/10/16/how-ai-mirrors-our-biases/
https://archimedes-lab.org/wp-content/uploads/2026/10/Pick-a-number-from-1-to-10-1024x784.jpg
#largelanguage #human #languagemodels #ai
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