“Since the beginning of #Israel’s war on #Gaza, academics in fields including #politics, #sociology, Japanese #literature, public #health, Latin American and Caribbean studies, Middle East and African studies, #mathematics, #education, and more have been fired, suspended, or removed from the classroom for pro-#Palestine, anti-Israel speech.”
I have a question about the aperiodic spectre tile (or the hat/turtle).
I know that the proof of aperiodicity works by showing that the tiles must fit together in a hierarchical structure that eventually repeats itself at a larger scale. But the larger units aren't literally scaled copies of the spectre. I also know that there is some freedom as to how you draw the edges of the spectre.
Is there a way you can draw the edges that allows you to literally use spectres to cover a larger copy of themselves? If so, is this way of doing it unique?
@OscarCunningham I'm pretty sure you can transform the hats' HTPF metatile system into a form where each higher-order metatile exactly covers a set of metatiles of the next order down. (Use the 'converged' metatile shapes; use a non-overlapping version of the expansion rules; do some horrible limiting thing that fractalises all the metatile edges.)
But then you still have four different fractally-shaped metatiles, and no way to decompose those into individual hats that are all congruent.
@OscarCunningham in fact, here's the paper I vaguely remembered seeing but couldn't put my hands on yesterday, which does pretty much what I said. https://arxiv.org/abs/2305.05639, diagrams on pages 7 and 8.
"Mathematics must subdue the flights of our reason; they are the staff of the blind; no one can take a step without them; and to them and experience is due all that is certain in physics." – Voltaire (1694-1778) #quote#mathematics#math#maths
Mathematician James Harris Simons, known for the classification of holonomy of 3D manifolds and his famous Chern-Simons form, passed away on May 10, 2024, in New York City, at age 86.
Despite his later cooperation with NSA to help US to invade Vietnam and entering financial business (which is notorious for redistributing wealth to enlarge economic inequality), his legacy in #mathematics and #physics still benefits our exploration in secrets of the universe.
Maths/CogSci/MathPsych lazyweb: Are there any algebras in which you have subtraction but don't have negative values? Pointers appreciated. I am hoping that the abstract maths might shed some light on a problem in cognitive modelling.
The context is that I am interested in formal models of cognitive representations and I want to represent things (e.g. cats), don't believe that we should be able to represent negated things (i.e. I don't think it should be able to represent anti-cats), but it makes sense to subtract representations (e.g. remove the representation of a cat from the representation of a cat and a dog, leaving only the representation of the dog).
@RossGayler@Heterokromia@cogsci to me it seems you need to be more clear on your requirements. Are your non-negative and multidimensional requirements independent, as far as you can tell?
If so, a multidimensional (do you know how many dimensions/animals you have?) modulo space sounds a viable solution. That'd be something denoted as https://www.HostMath.com/Show.aspx?Code=%5Cmathbb%7BZ%7D_k%5En , with k being the cardinality of one dimension (would they need to have different cardinalities?), and n being the number of dimensions.
Why are algorithms called algorithms? A brief history of the Persian polymath you’ve likely never heard of.
Over 1,000 years before the internet and smartphone apps, Persian scientist and polymath Muhammad ibn Mūsā al-Khwārizmī invented the concept of algorithms.
I bet that a lot of people in the Fediverse already know this very pretty pencil-based 3D art. But in case you haven’t, be prepared to marvel.
This sculpture is known as the hexastix and a variant series created by artist George Hart is titled 72 Pencils.
If you can get 72 unsharpened hexagonal pencils, and some flat rubber bands, you can attempt to create this. Search for a video by @standupmaths for a pseudo-tutorial.
"Numbers are free creations of the human mind, they serve as a means of apprehending more easily and more sharply the diversity of things." – Richard Dedekind (1831-1916) #quote#mathematics#math#maths#numbers