Why Gauss wanted a heptadecagon on his tombstone
Mathematician Johann Carl Friedrich Gauss proved the constructibility of a heptadecagon at 18, solving a long-standing problem. His legacy is symbolized by a 17-pointed star in his birthplace.
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Mathematician Johann Carl Friedrich Gauss, known for his significant contributions to mathematics, particularly in geometry, desired a regular heptadecagon (17-sided polygon) to be etched on his tombstone. This shape represented a major achievement in his work, as he proved its constructibility using only a compass and straightedge, a problem that had perplexed mathematicians for over 2,000 years. Gauss's proof, completed at the age of 18, demonstrated that the length required to construct a heptadecagon could be expressed using basic algebraic operations. This breakthrough not only established the heptadecagon's constructibility but also led to a broader understanding of which regular polygons can be constructed with these tools. Gauss's work was later validated by Pierre Wantzel, who confirmed the impossibility of constructing certain polygons, including the heptagon and hendecagon, with a compass and straightedge alone. Although Gauss's wish for a heptadecagon on his tombstone was not fulfilled, a monument in his birthplace features a 17-pointed star, symbolizing his legacy in mathematics.
- Gauss wanted a heptadecagon on his tombstone to symbolize his mathematical achievements.
- He proved the constructibility of the heptadecagon at age 18, solving a problem that had stumped mathematicians for centuries.
- His work established criteria for which regular polygons can be constructed with a compass and straightedge.
- Pierre Wantzel later confirmed Gauss's findings regarding the impossibility of constructing certain polygons.
- A monument in Gauss's birthplace features a 17-pointed star in lieu of the heptadecagon.
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- Many commenters express a desire for more recognition of Middle Ages mathematicians, contrasting them with Greek and Renaissance figures.
- There is curiosity about the limitations of compass-and-straightedge constructions, particularly regarding polygons with odd numbers of sides.
- Some users share resources, such as videos and links, to further explore Gauss's proof and related mathematical concepts.
- Discussion includes the aesthetic and practical aspects of geometric constructions, with recommendations for hands-on practice.
- Questions arise about the symbolism of Gauss's legacy, including the presence of a 17-pointed star on his headstone.
15 commentsDetailed description of the problem of constructable regular polygons and a gloss of the proof. https://www.youtube.com/watch?v=EX7U0DGBmbM
A full explanation of the proof: https://www.youtube.com/watch?v=Gdy1u4lsjDw
A length is constructible exactly when it can be expressed with the operations of addition, subtraction, multiplication, division or square roots applied to integers.
[...]
Remarkably, the rudimentary tools that the ancient Greeks used to draw their geometric diagrams perfectly match the natural operations of modern-day algebra: addition (+), subtraction (–), multiplication (x), division (/) and taking square roots (√).
The reason stems from the fact that the
equations for lines and circles only use these five operations
, a perspective that Euclid couldn’t have envisioned in the prealgebra age."
Related:
Oliver Byrne made an insanely pretty colourful version of Euclid's Elements, which is available online. Grab a pen, paper, a string to make circles, and the edge of a book to draw straight lines, start with Proposition 1 and go as far as you'd like: https://www.c82.net/euclid/book1/#prop1
(There is also a physical facsimile of Byrne's Elements (ISBN:9783836577380) – it is one of the best additions I've ever made to my library. It is simply gorgeous.)
You can't do it exactly, but you can do it to arbitrary degrees of accuracy; at least about as far as you can go without bumping into the precision limits of a compass and straightedge.
1/7 = 1/8 + 1/64 + 1/512 + 1/4096 + 1/32768... as you can see this will hit the limits of human precision in short order.
In general any fraction 1/(2^n - 1) can be expressed as an infinite sum (or a series that comes infinitesimally close)
1/(2^n - 1) = the sum from x equals one to infinity of 1/(2 ^ (x * n)). And we all know how to section any arch-length into fractions over powers of 2.
So starting with a complete loop, segment take the first piece, then take the second piece, segment and take its first piece... keep on adding all the little pieces together until it's so close enough to 1/7 that you can take a compass measure and use that to resegment the rest of the pie - making sure that you recurse enough that after you've market out 6 additional ones, you get near enough a collision to the first that you're not really worried.
But yeah, I'd be surprised if you could compass and straightedge even to a precision of one part in 4096 - and there's no way in hell that anyone's ever pulling off one part in 32768.
Everytime I hear about Yitang Zhang, I cannot help but be amazed by his accomplishment.
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