This month’s topics:
Old Babylonian Applied Geometry
The story was picked up by The Guardian, the Smithsonian’s SmartNews, Science News, Arab News and Popular Mechanics. Donna Lu’s coverage in The Guardian (August 4, 2021) has the headline “Australian mathematician discovers applied geometry engraved on 3,700-year-old tablet”, with subhead “Old Babylonian tablet likely used for surveying uses Pythagorean triples at least 1,000 years before Pythagoras.”
Lu begins: “Known as Si.427, the tablet bears a field plan measuring the boundaries of some land. The tablet dates from the Old Babylonian period between 1900 and 1600 BCE and was discovered in the late 19th century in what is now Iraq. It had been housed in the Istanbul Archaeological Museum before Dr Daniel Mansfield from the University of New South Wales tracked it down.”

the Istanbul Archaeological Museum, courtesy of Daniel Mansfield.
First note that the area surveyed in the tablet has been partitioned into simple geometrical shapes (rectangles, right triangles, trapezoids) for which the Babylonians knew how to calculate the area from the linear dimensions.

Red: lengths in rods. Blue: areas in sar, iku, eše.
Mansfield noticed the presence of two or maybe three Pythagorean triangles (right triangles with whole-number sides) in the diagram, including the triangle
Computer assists lofty theory
Davide Castelvecchi’s contribution to Nature for June 18, 2021 was “Mathematicians welcome computer-assisted proof in ‘grand unification’ theory,” with sub-head “Proof-assistant software handles an abstract concept at the cutting edge of research, revealing a bigger role for software in mathematics.” He tells us how Peter Scholze (Bonn; “considered one of mathematics’ brightest stars and has a track record of introducing revolutionary concepts”) and his collaborator Dustin Clausen (Copenhagen) have set forth an “ambitious plan,” which they call condensed mathematics: “they say it promises to bring new insights and connections between fields ranging from geometry to number theory. … Until now, much of that vision rested on a technical proof so involved that even Scholze and Clausen couldn’t be sure it was correct. But earlier this month, Scholze announced that a project to check the heart of the proof using specialized computer software had been successful.”
The specialized software in question is a proof assistant. Castelvecchi: “Proof assistants … force the user to lay out the logic of their arguments in a rigorous way, and they fill in simpler steps that human mathematicians had consciously or unconsciously skipped. … In this way, proof assistants can help to verify mathematical proofs that would otherwise be time-consuming and difficult, perhaps even practically impossible, for a human to check.”
Castelvecchi goes on to tell us something about condensed mathematics, and the ensuing ‘grand unification’ project; finally: “Around 2018, Scholze and Clausen began to realize that the conventional approach to the concept of topology led to incompatibilities between [the] three mathematical universes — geometry, functional analysis and
“There was one catch, however: to show that geometry fits into this picture, Scholze and Clausen had to prove one highly technical theorem about the set of ordinary real numbers.” Scholze found a proof, but is was so novel and complex “that Scholze himself worried there could be some subtle gap that invalidated the whole enterprise.” Here is where Kevin Buzzard (Imperial College, London) and Johan Commelin (Freiburg), along with a team of other experts in the proof-assistant package Lean, enter the picture. “By early June, the team had fully translated the heart of Scholze’s proof — the part that worried him the most — into Lean. And it all checked out — the software was able to verify this part of the proof.”

Fractals and Synchronization
On June 18, 2021, SciTechDaily posted “A New Bridge Between the Geometry of Fractals and the Dynamics of Partial Synchronization”, a press release from Universitat Pompeu Fabra, Barcelona. It refers to “Chimeras confined in fractal boundaries in the complex plane,” published in Chaos May 3, 2021 by UPF Professor Ralph Andrzejak. “The work generalizes the Mandelbrot set for four quadratic equations.”
The Mandelbrot set comes up in the study of the iterates of the complex function function
Andrzejak studies four coupled copies of the iteration, with the same value of
The behavior of the network depends, delicately of course, on
Color | Type | Formula |
Light Blue | Full synchronization | |
Orange | Within-pair synchronization | |
Green | Across-pair synchronization | |
Magenta | Chimera state | |
Red | Full desynchronization |

The SciTechDaily posting ends with the question, “whether the mathematical model in question can be relevant to the dynamics of the real world.” They give Andrzejak’s answer: “Yes. Absolutely. The best example is the brain. … Our brain can only work properly if some neurons synchronize while other neurons remain out of sync. … If we study the basic mechanisms of partial synchronization in very simple models, this can help understand how it is established and how it can be kept stable in such complex systems as the human brain.”