Saturday, February 7, 2026
Russia launches Friday morning rush hour missile, drone attack on Ukraine -- Drone discovered on Moldovan territory near Ukraine border -- 18 prisoners in 24 hours: Ukraine's Azov International reveals video of Dobropillia clearing operation -- Canadian female fighter at the forefront of Ukraine's drone war ... and morehttps://activitypub.writeworks.uk/2026/02/saturday-february-7-2026/
This Geometry Puzzle Took Awhile!
Illustration by Charles Hinton, from The Fourth Dimension (1904).
Source: University of Toronto Libraries / Internet Archive
https://pdimagearchive.org/images/98c47b7a-4c26-4b4b-9fe2-a04d8d9aa24c
#mathematics #spacetime #cubes #physics #geometry #tesseracts #art #publicdomain
According to the biography by Diogenes Laertius, Pythagoras (c.570–c.490 BCE) ‘held that the most beautiful figure is the sphere among solids, and the circle among plane figures’.
This aesthetic preference for the circle and sphere can be traced through thinkers like Plato (who, according to later writers, set the problem of describing the movements of the heavens using uniform circular motions), Cicero (106–43 BCE), and Proclus (410/12–485 CE), and into the middle ages.
Thomas Bradwardine (1290/1300–1349), one of the mediaeval ‘Oxford calculators’, was obviously influenced by this tradition when he wrote that the circle ‘is the first and most perfect of figures, the simplest and most regular, the most capacious and the most beautiful of figures’.
But Bradwardine then presented evidence that he saw as attesting to the beauty and perfection of the circle: (1) the construction to find the centre of a circle by bisecting a diameter found as the perpendicular bisector of a chord; (2) that the intersections of six equally-spaced radii with the circumference define a regular hexagon; (3) that exactly six circles of equal size can touch a given circle (see attached image).
For Bradwardine, the perfection of the circle was thus linked to the perfection of the number 6 = 1+2+3: the construction involves six intersections with the circle; the hexagon is made up of six lines; the third result involves six outer circles.
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#MathematicalBeauty #HistMath #Pythagoras #Bradwardine #geometry #aesthetics #PerfectNumber
Module 2 of 2.
Two ways of cutting an infinite 3-periodic heptagonal tiling along higher order modules.
Module 1 of 2.
#inkscape is still working perfectly with the Brother #scanncut. No real software/format issues which is surprising and pleasant. (Although to re-iterate, text needs to be converted to path. But that's true in (almost?) all vector graphics-processing toolchains)
That said, my particular project is having some hiccups
Workflow: My client/wife wants #Galentines shotglasses. I made a single "blank" and then added alternate designs as layers. But when you load that in the machine all the layers are still obviously there right on top of each other
I messed a little Inkscape's "export only selected layers" but that didn't seem to do what I expected for unclear reasons. In any case, it really makes more sense to have alternate designs spread around a page to view all at once. That makes them easier to view and is how I'll want to print the final job anyway.
#3D #Geometry: The shotglasses are heart-shaped and also have some curvature from top to bottom. There's just no way to wrap a flat piece of vinyl around a positive-curvature item without wrinkles.
I think I've got the design small enough and with gaps in the right places that I can make it work. It also helps a lot to remove the transfer sheet early and just lightly deform only the vinyl itself until it is flat.
I mocked up one shotglass last night as a design review for the client but haven't heard back. Final product is due on Tuesday.
A New AI Math Startup Just Cracked 4 Previously Unsolved Problems
As noted in a previous post, Archimedes thought highly of the result that the ratio of either the volumes or surface areas of a cone, a sphere, and a cylinder exactly circumscribing them is $1:2:3$.
So did others: three centuries later, the architect Nicon (d.149/50 CE), father of the philosopher and physician Galen (129–c.210/217 CE), thought it fitting to point out the ratio of the configuration in a public inscription in his city, Pergamon:
‘the cone, the sphere, the cylinder.
If a cylinder encloses the other two shapes,
[...]
Competition the principle and in solids
the progression $1 ∶ 2 ∶ 3$,
a noble, divine equalization,
but also mutual interdependence
of the solids, always in the ratio $1 ∶ 2 ∶ 3$.
They should be beautiful and wonderful,
the three solid shapes’
Nicon doubtless admired these ratios as an architect: a sphere inside a cylinder brings to mind the Pantheon at Rome, of which the Temple of Zeus Asclepius Soter in Pergamon was a half-scale copy. These buildings were designed so that a basically cylindrical rotunda was crowned with a hemispherical dome under which a sphere would fit (see attached image).
[Each day of February, I am posting a story/image/fact/anecdote related to the aesthetics of mathematics.]
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#MathematicalBeauty #HistMath #Archimedes #geometry #architecture #aesthetics
I've become interested in polyominoes recently and have had a few questions about them I couldn't easily find answers to. Here's one:
Let's call an n-omino "balanced" if its 2 colouring has \(\lfloor\frac{n}{2}\rfloor\) cells of one colour and \(\lceil\frac{n}{2}\rceil\) cells of the other colour. Is there a nice structural characterisation of balanced n-ominoes?
I am looking at tilings of a certain kind and I can rule out unbalanced n-ominoes and non-trees. So I am particularly interested in what balanced trees look like.
Follow up: is every 2-connected n-omino balanced?
How can angle BPC (labelled x) be found using basic circle theorems/geometry?
Do I need some construction lines to unlock the problem, perhaps through the use of similar triangles APC and BPD?
Thank you in advance for any help offered.
Detail of a fountain at the Hassan II Mosque, Casablanca, Morocco
#TilingTuesday #geometry #tiling #MathArt #photography #design #IslamicPattern #IslamicArt #zellij #zellige #TravelPhotography
Max Dehn (1878–1952) said that Archimedes’ (c.287–212 BCE) discovery that the surface area of a sphere was four times its great circle was the one of the most beautiful results of Greek mathematics.
Archimedes himself had a high opinion of this result and two others in his two books ‘On the Sphere and the Cylinder’: that the volume and surface area of a sphere and a cylinder exactly circumscribing it are in the ratio $2 : 3$. One can add a cone fitting inside the cylinder to have ratios $1 : 2 : 3$ (see 1st attached image).
It has been suggested that Archimedes’ conjectures for these ratios may have been guided by a conscious or unconscious search for beautiful integer ratios between geometric configurations. There is no direct evidence for this motivation, but Archimedes’ work seems to exhibit a preference for small integer ratios.
According to Plutarch, Archimedes desired that his tomb should be marked by a cylinder enclosing a sphere and an inscription of the ratio of the one to the other; Cicero related how he had sought out Archimedes’ tomb and found a column just so inscribed (see 2nd attached image).
[Each day of February, I intend to post an interesting story/image/fact/anecdote related to the aesthetics of mathematics.]
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#MathematicalBeauty #HistMath #Archimedes #Plutarch #Cicero #geometry #aesthetics
