The calculation of the number pi has been pursued for thousands of years, using both geometric and infinite series techniques. For example, the Leibniz series for pi is 4 times the infinite series (1 β 1/3 + 1/5 β 1/7 + 1/9 β¦). Since the advent of modern computers, pi has now been calculated to as many as 31.4 trillion digits [1]. However, there are also mechanical methods to calculate pi, one of the earliest being the method of Georges Louis Leclerc Comte de Buffon [2], in which a needle of length L/2 is dropped at random many times onto a grid of parallel lines a distance L apart; for a large number of trials, the fraction of times the needle intersects a line is approximately 1/pi.

More recently, in 2003, Gregory Galperin showed that two masses colliding with each other and with a fixed wall may be used to calculate pi to any desired accuracy if the collisions are elastic, i.e., no energy is lost to friction [3]. In his construction, a large mass collides with a small mass at rest. The masses are constrained to move in one dimension, so that when the smaller mass then hits the wall, it bounces back and hits the larger mass again. This process continues until the larger mass reverses direction and is moving away from the smaller mass faster than the smaller mass is moving, and there are no more collisions. If the ratio of the masses is 100 raised to the *k*th power, then pi is approximately the number of collisions (between the masses and between the small mass and the wall combined) divided by 10 raised to the *k*th power. For example, if the ratio of the masses is 10,000:1, there will be 314 total collisions and the approximation for pi is 3.14.

One key to this remarkable situation is that the elasticity of the collisions ensures that the total energy is conserved (more on this later). However, when the smaller mass rebounds off the fixed wall — which is being treated as an immovable object that has infinite mass — total momentum is not conserved. An infinitely massive wall is not physically possible, and it is a fundamental law of physics that momentum is conserved. I wondered if you could come up with a process to calculate pi that did not require a fixed wall and therefore would also conserve momentum.

I found such a process. Start with *three* masses constrained to move in one dimension and colliding elastically, with the mass on the left moving and the other two initially at rest (see diagram below). Let the first and third masses be equal and much larger than the middle mass; then the number of collisions *N* in this situation will also lead to a good approximation of pi.

In this case, if *r* is the ratio of the masses, pi will be given approximately by N times the square root of 2/*r*. For example, if the ratio *r* is 20,000:1, then there will again be 314 collisions and pi is approximately 3.14. It is interesting that with two large masses the mass ratio has to be twice as much as in the situation with one large mass and a wall.

Why are conservation of energy and momentum somehow related to pi? Conservation of energy can be written as a quadratic equation in the three velocities, which defines an ellipsoid in the three-dimensional velocity space β sort of like a football, even more like an Australian rules football that is more rounded at the ends. Conservation of momentum is an equation linear in the velocities, which defines a flat plane in velocity space. The system is constrained to lie on the intersection of these two surfaces, which is an ellipse. By stretching the coordinates appropriately, the ellipse can be changed into a circle. The remarkable thing is that each collision advances you by a set amount around this circle, and for very large *r *you have to go almost exactly halfway around the circle to reach the point where the masses no longer collide with each other. And, of course, halfway around a circle is pi radians, so the number of collisions leads to a good approximation for pi when *r* is very large.

Can you come up with other scenarios to approximate pi, perhaps with more masses? Maybe so, but it is much trickier. In the two cases above, the smaller mass is always bouncing between a large mass and either a wall or another large mass. The order of the collisions is always the same β the smaller mass always alternates colliding with something on its left and something on its right, regardless of the exact initial positions. With more masses, the first mass hits the second, which hits the third, but whether the next collision is the second mass hitting the first again or the third mass hitting the fourth mass will depend on the relative initial positions as well as the mass ratios. Iβm content with sticking with the three-mass solution.

An appropriate song, especially at the 0:16 mark.

[1] See the article by E. Haruka Iwao, βPi in the sky: Calculating a record-breaking 31.4 trillion digits of Archimedesβ constantβ on Google Cloud, https://cloud.google.com/blog/products/compute/calculating-31-4-trillion-digits-of-archimedes-constant-on-google-cloud .

[2] G.L.L. Comte de Buffon, βSur lej eu de franc-carreauβ (1777). That this gives an approximation to pi can be proved using calculus.

[3] G. Galperin, Rational and Chaotic Dynamics, Vol. 8, No. 4, 2003, p. 375. A nice discussion of this situation can be found at https://www.youtube.com/watch?v=jsYwFizhncE .

I liked the needle drop method the most

All of the mechanical methods are fascinating.

Yes, the mechanical methods are perhaps unexpected. The needle drop has the issue that it will have a statistical uncertainty that requires a large number of trials to beat down. The colliding blocks require an extremely large mass ratio just to get a few significant digits — not to mention requiring no friction.

Assume a frictionless cow.

If a frictionless cow has a non-aerodynamic shape, would it still suffer from drag as it forces air out of its flight path?

Yes, since drag is a result of forces besides friction on the surface (in particular, the difference in pressure in the fluid on different parts of the body).

STEVE SMITH WANT SEE THIS FRICTIONLESS COW.

HIM HAVE REASONS.

I was Math Team MVP for two years in high school. Unfortunately, it’s not like riding a bike.

I lettered in Math. I was a mathlete!

I was a Mathlete. Mrs. Prole thought it was hilarious that that was a thing.

This x1000. Math beyond basic algebra and geometry that we do without thinking (because it is in the world of basic living) needs to be done nearly daily to be proficient.

I have forgotton more Calculus than most people will ever know.

Same.

And I wouldn’t care except I enjoyed it so much at the time.

Unlike all the other crap I forgot like biology or social studies or or or

Math was so satisfying. It was true, regardless of what the world around you told you. 2+2 = 4. No other fact. You could prove it in the real world. I miss math really. I have at one point actually done some online stuff going through math ‘classes’ just to exercise that itch.

A=A?

When I went to college I deliberately rejected everything I was good at. Math, science, computers.

It took me another decade to recover from that stupidity and learn myself into a software career that I quite like.

Why?

Dammit, I misremembered “A is A”.

Not gonna speak for rhywun, but I would assume its because it is known. Rather than exercise that portion of the brain and be the driving force in what might be a career, you put it aside and explore other options thinking it was mastered. At least, that is how my brain would see that.

I don’t know why.

And I greatly regret it. But oh wellz.

Uuuuuuuuuhhhh…. So. boring.

Math isn’t hard, but just boring as all get out.

I’d taken the minimum number of math courses I could in both high school and as an undergrad. Some years back I had delusions of pursuing some sort of graduate business degree and enrolled in a Business Math course. Imagine my terror when I found the textbook had the “C” word on the cover. Fortunately, I’d bought the book early, and it had a review of algebra in front. I plowed through that review as fast as I could. Somehow managed to ace that class and for some reason decided to keep the book. Every so often I pull it out and go through some of the algebra review again…but I’ve never ventured back into the “C” stuff.

I joined the Math Club as a FU to HS life, but wasn’t good at it at all.

In my small HS, no one wanted to join the math club. So they made everyone in the HS take a math test, the highest scores were forced into math club. Even feigning stupidity I was too smart. I tried to fail the test, but my 20 out of 100 put me in the high scores.

In Physics 1, one of the labs was to measure the value of Pi. The actual lab was about error margins.

My team calculated Pi to be 6Β±3. Technically Pi fell within that error margin.

Your 8 figure global warming grant, sir.

+1 Indiana Pi Bill

LOL.

I vaguely remember the 1979 version of the Guinness Book of World Records – before they dumbed it down – had a record for “lamest approximation of pi” or some such.

The value was “4”.

Mathematicians be weird, yo.

The Deaner gets it.

I’ll take the weird idiosyncrasies of math geeks over the smarmy shit you get from lawyers any day of the week.

*eyes glaze over*

*makes a stiff drink*

I hope you meant, makes another stiff drink.

“Stiff drinks” might have something to do with why I don’t math anymore.

Yeah. I was reading the method for calculating pi using needle drops to the wife, and she told me to stop it. I thought it was exciting.

I think the needle drop method is cool, and it’s neat that you can prove it with calculus. I think the reason pi comes from that is related to the fact that randomly dropping a needle involves the angle of the needle having a range of 0 to 2 times pi. although it’s not quite as direct as the for the colliding masses where the system is clearly going halfway around a circle in velocity space.

Whoa, dude, What?

Give him ten more minutes.

For the drink to thaw?

To pour another drink.

Now can we prove that with math?

PI(avg responses to post x hours since post published in minutes) / your time to comprehend in minutes = minutes between pours

3.14(22 * 80) / 80

3.14(1760) / 80

5526.4 / 80

69.08 minutes between pours….which…checks out.

Russel’s paradox. Those darned stubborn analogies.

Math is the square peg that the corners just shave off a little bit upon driving in.

Truth and complete understanding can only be approached asymptotically. They shall never meet.

The important question is how to measure the distance between the two.

Does that test work for everything, the abstract and the subjective?

Inquiring minds want to know.

Wouldn’t a simpler “mechanical” method be to measure the circumference of some large circular object and divide by the diameter, using some string and a tape measure?

That was the lab we did in Physics, Either that or we drew a circle with a compass and tried to encircle it with a string to get that measure. Something along those lines with the string method.

Alternatively if you had a cylinder, you could mark the rim and a paper under it, roll it until the mark was at the bottom again, mark that point, measure between marks.

Mind you, I don’t need a large number of significant digits.

Now you’re just being ridiculous.

That works, but only to the accuracy of your measuring device. Infinite series methods or the mechanical methods mentioned here in principle can get as accurate as you want.

Here is the question I’ve never seen brought up… is there any reason to know Pi to 1 trillion digits?

Not yet.

Firstly, I’m just going to divide all those numbers by zero, then declare winning.

careful with that

We had a major outage because of code in the software that tried to divide a negative number by zero.

Why they were dividing by zero in the first place is beyond me but the fact that some engineer didn’t account that someone in the field might put out a negative number is even more baffling. It is the “why would they do that?” thinking.

For context, it is altitude encoding of a transponder and the system would just zero them out and use a hard-coded number for each known transponder (our test transponders really). Well, the military was running the transponder up and down in altitude and took it below sea level by a 1000 ft. System just shit the bed.

“We needed to make a water landing on the Dead Sea.”

The engineer who wrote the code was quite embarrassed when I called him about it and he debugged what happened. He kicked himself for quite sometime.

Perrrrfect.

That is a water intake for the dam. π

@OBE: Were these getting used on subs?

I had an awesome 9th grade physics teacher. We had great fun shocking the last person in the chain with the Van de Graff generator.

His final test was to approximate the velocity of a BB pellet out of a BB gun using a ballistic pendulum. OK, OK, we may have also shot out a few of the gym lights, too, but it was for SCIENCE!

One of my favorite demos when teaching is the bed of nails. Another is the bowling ball pendulum, where you swing it away from your face and it comes back to almost hit you.

I had a co-worker who was in a bad accident in college. During his recovery, he memorized pi to 100 digits.

To this day, he sends me an email every March 14th.

How many digits is he up to now?

In Chinese mythology, my personhood was predicted as Ming The First.

Just take the Indiana approach!

https://en.m.wikipedia.org/wiki/Indiana_Pi_Bill

A hundred years later and they’re still trying to pass pie bills

https://www.winchesternewsgazette.com/news/legislature-to-consider-designating-sugar-cream—hoosier-pie/article_b1c4e6d2-e761-591e-bbad-eb7d499d7c3b.html

Better approximations of Ο than those implied by the bill have been known since ancient timesyour government at work; progressive impulses; etc

Leon Russell comments on pi

https://www.youtube.com/watch?v=pwxs7N9cwcY&list=PLAKBCdHlMy4_iuwGvpMrHkRJ-JXYWrOKr&index=10

I appreciate you explaining why momentum can encode for Pi in this construction. Often this type of calculation is presented without any reason why it has a relation to circles.

Happy hump day!

ππ

https://m.youtube.com/watch?v=LXEKuttVRIo

Excellent song. Long video intro though. πΆπΆ

Morning, Sean.

βπ

Morning Glibs.

Good morning all!

Today we have tunes from John Foxx, post-Ultravox.

He’s a Liquid.

Underpass.

Share and enjoy!

Good morning, Beau, U, Sean, and Lack!

Coffee’s almost ready…

YT helpfully segued right from the first to the second, and now is feeding me some lovely instrumentals from his “Arcades Project.” Very nice! Thanks!

He’s done quite a varied catalog of work, from the ‘hard electronica’ of his first solo albums to ambient electronica in later years. It’s an interesting body of work. More to come!

Meat grinder arrived overnight, so I have to make the counter photography clean not just livably clean. I Think I’ll document my attempts at making sausage for an article. See what happens.

As long as you don’t show us how legislation is made. π«

well played

Thank you, Don, and good morning.

Aww…

On U’s PC screen:

~~“I’m Just a Bill”~~Actually, it’s just my email and a few browser windows – a 3d printer info page, a ukranian artist’s gallery, glibs, and a cooking show.

I don’t start drafting articles that need pictures until I’ve taken pictures. And my cooking articles are written after I did the test cook.

Hope the artist’s gallery is well back from the front. Worth sharing?

(Or is it a “galleryy”?)

Don’t know where in the country he is, but he’s still posting

https://www.deviantart.com/lipatov/gallery

One of my favorites from the collection: https://www.deviantart.com/lipatov/art/Kaiju-187856375

Impressive!

Previous comment was regarding the collection as a whole. That individual one is cute! π

There’s a reason I follow his work.

Alternate response – Dat’s Plagarisms!

When I see the word ‘Reform’ bantered around legislation I immediately see “We screwed up the first time”

i.e. Education reform, banking reform, etc. An admission of failure…

Do we still have reform schools? With room for 535 new students?

Need. More. Coffee.

Morning all.

Good morning, Suthen!

Need moar SpO2.

Good morning, GT & Suthen!

Good morning, hayek! I wish you easy breathing!

Mornin’. Better breathing ahead, I hope.

Good morning, good morning, the best to everyone his morning. Good to see everyone here!

Good morning, 4(20)!

See? I can do SOME math!

Mornin’, reprobates!

Math and morning Covfefe do not mix.

See Mojeaux’s link from yesterday’s midday post re: the molecular structure of caffeine.

That would be SCIENCE!!!11!1!!

Too early for that as well.

No math. Iβm off duty. At least for 15 more minutes. A long, luxurious morning of dish washing and coffee making.

For the metal heads: https://www.youtube.com/watch?v=Wl_11NkhD-g

This guy’s songs are amazing.

Yum. I prepared to make some biscuits and tomato gravy but Mrs. Suthenboy ran me out of the kitchen. She wanted to do it.

I knew I had a good reason for marrying her. Good God that is delicious.

Since we seem to be doing music in the mornings lately: https://www.youtube.com/watch?v=_Cpu9Z3z6pU

A morning without music is like a morning without sunrise.

Tomato gravy?? Is that a Cajun thing?

Oh my God. You are kidding, right?

2 tablespoons butter or bacon grease/lard

1/2 small yellow onion, finely chopped

1 clove garlic chopped finely

2 tablespoons all-purpose flour

1 cup chicken broth

We used 1can diced fire-roasted tomatoes

If the broth had no salt then salt to taste

1-2 teaspoon sugar or Splenda

Make your roux with the chopped onion in it (butter or fat, flour, onion, garlic)

cook until onion is clear

Add broth, tomatoes, salt if needed, and sugar.

Cook down to halfish

You can of course add in other spice…some like rosemary or thyme

I like some basil or parsley

Split your biscuits and douse generously

Try not to eat too much