Some math fan service

by liberal japonicus

The discussion of programming brought a lot of math love out, so I thought I'd toss this up as a quasi-open thread.

This Guardian article just popped up about two high school students presenting about a trigonometric proof of the Pythagorean theorem. I'm still wondering what the shape of the proof would be, but the internet tossed this interesting page up, a list of all the proofs of the Pythagorean theorum, part of a larger labor of love site called Cutting the Knot by Alexander Bogomolny. Bogomolny passed away in 2018, but the site is still being curated. It's pretty massive, so anyone who pokes around in it, if you have a page or section you like, bring it up here, toss in your thoughts about what a trigonometric proof of the theorem would be or give any other math things you feel like putting up.

 

Comments

[wj]

Ramadan Kareem! to you all.

[Pro Bono]

Proof #4 is my favourite.

[nous]

Byrne's Euclid, hosted by the math department:

https://www.math.uci.edu/~ndonalds/Elements-I-VI.pdf

Book 1 Proposition 47.

I remember it well from the freshman math tutorial at St. John's College (Santa Fé).

[nous]

Byrne's Euclid is a thing of beauty, adding a new level of aesthetic and intellectual delight to the already delightful Elements. It's one of those books I might get just to have on the shelf even though I doubt I will ever go through the thing again.

I gave away my copy of the Elements from the Britannica series a few years back when we had to downsize apartments. Think that's when I unloaded Ptolemy and Aquinas as well.

[Pro Bono]

Put a square of side 'c' at an angle inside a larger square, such that the corners of the smaller square touch the sides of the larger square.

Let the distances between the corners of the larger square and the corners of the smaller square be 'a' and 'b' - by symmetry this is the same on all four sides. The larger square then has sides of length a+b.

The area of the larger square is the area of the smaller square plus the areas of four right-angled triangles each with hypotenuse 'c' and other two sides 'a' and 'b'.

That is:
(a+b)^2 = c^2 + 4.ab/2 = c^2 + 2ab
a^2 + b^2 + 2ab = c^2 + 2ab
a^2 + b^2 = c^2

[CharlesWT]

Put a square of side 'c' at an angle inside a larger square, ...

"There is no error in this statement. It correctly derives the Pythagorean theorem using a geometric approach with squares and right-angled triangles." —ChatGPT

:)

[hairshirthedonist]

I posted this in the wrong thread, thus the lead-in. I should have known that the other comments weren't mathy for a very good reason.

Something more mathy:

https://www.wired.com/story/unbelievable-zombie-comeback-analog-computing/

I just happened to come upon that article after watching this episode of Nova last night:

https://www.pbs.org/wgbh/nova/video/einsteins-quantum-riddle/

Later in the program, they got into the subject of quantum computing, which was described as being more "fuzzy" than traditional digital computing. To me, that sounds at least analog-ish. So I was expecting the zombie-analog article to be about quantum computing. Apparently not.

I don't have time to write a whole lot about this right now, but I know there are people here who know a lot more about such things than I do, so maybe a starting point for further discussion.

[Michael Cain]

Somewhat related to analog computing, digital circuits running faster than about 20 MHz often require careful analog considerations of the layout. At those frequencies printed circuit board traces all act as waveguides. If you put a right-angle corner in a waveguide, Bad Things happen. (Contemporary layout software takes care of this automatically.) As frequencies increase farther, signal propogation time comes into play so PCB bus traces must be close enough to the same length that propogation times in all the runs are nearly the same. Parasitic capacitance rears its ugly head in a variety of ways. Power distribution design is a headache.

Fortunately I never had to design boards that ran above 20 MHz, so I could pretty blindly follow a few rules of thumb to avoid the analog problems.

[Snarki, child of Loki]

20MHz isn't that big a deal.
The fun starts around 100MHz, and gets really crazy upwards of 1GHz.

[CharlesWT]

The computational lithography used to design the masks for a chip is so computationally complex that it can take a whole data center a couple of weeks to do the computations for a single chip. But the process has been sped up by about 40x using GPU processors. It surprises me that they're only doing this now since GPUs have for some time been used for a number of applications other than the one they were originally intended for.

"In this Video I discuss New Breakthrough in Chip Manufacturing which speeds up Computational Lithography by 40 times!"
Breakthrough in Chip Manufacturing - x40 times faster

[CharlesWT]

Provide an overview of computational lithography.

"Computational lithography is a field of study that focuses on creating algorithms and software tools that enable the design and manufacture of smaller, faster, and more complex integrated circuits. It involves the use of mathematical models and simulations to optimize the lithography process, which is responsible for printing patterns on the surfaces of silicon wafers. Computational lithography plays a critical role in the semiconductor industry by enabling the production of smaller and more powerful microchips, which are essential components in electronic devices such as smartphones, computers, and cars. It involves the integration of various technologies such as optical physics, computer science, and mathematics to develop sophisticated algorithms that can compensate for the limitations of lithographic systems. Computational lithography is constantly evolving, with researchers developing new methods and tools to keep up with the demands of the industry."

Describe the level of computational intensity required for computational lithography.

"In terms of computational intensity, computational lithography is a highly intensive process that requires a significant amount of computing power and resources. The computations involved in computational lithography are complex and require high levels of accuracy, which necessitates the use of large-scale computing systems such as parallel computing clusters and supercomputers. The process involves simulating the behavior of light as it interacts with the materials used in the lithography process, which requires solving complex mathematical equations and simulating the behavior of light at a high level of detail. The level of computational intensity required for computational lithography is therefore very high, and it is a key factor that limits the speed and efficiency of the lithography process." —ChatGPT 3.5 Turbo

[CharlesWT]

What areas of math are used in computational lithography?

"In terms of specific areas of math used in computational lithography, here are a few examples:

1. Fourier analysis: Fourier analysis is used to decompose a complex pattern into a series of simpler sinusoidal waves. This is useful in computational lithography as it allows for the optimization of the diffraction patterns produced in the lithography process.

2. Optimization algorithms: Various types of optimization algorithms are used in computational lithography to find the best possible solutions to complex problems. These algorithms include convex optimization, nonlinear optimization, and integer programming.

3. Partial differential equations: Partial differential equations are used to model the behavior of light as it interacts with the materials used in the lithography process. This allows for the accurate simulation of the lithography process and the optimization of the diffraction patterns produced.

4. Linear algebra: Linear algebra is used in computational lithography to solve systems of linear equations that arise in the modeling and simulation of the lithography process.

5. Numerical methods: Various numerical methods such as finite element methods, finite difference methods, and boundary element methods are used in computational lithography to solve complex mathematical problems that arise in the lithography process." —ChatGPT 3.5 Turbo