Billiards and Math

justnum

justnum

Billiards Improvement Research Projects Associate
Silver Member
For the billiards and "math/physics" people.

Wolfram alpha is a popular math website.

If you can't recruit people to play pool, it might be worth creating some math problems for the math community.
With all the youtubers promoting pool, it would be likely that the another level is to get the academic community involved.

Everyone can try to discover well known physics concepts, they just need to know the academic language that goes with it.

The billiards table is well equipped to introduce Physics concepts and some concepts about relative space.





https://mathworld.wolfram.com/Billiards.html
1639407223649.png

Given a rectangular billiard table with only corner pockets and sides of integer lengths
m
and
n
(with
m
and
n
relatively prime), a ball sent at a
45 degrees
angle from a corner will be pocketed in another corner after
m+n-2
bounces (Steinhaus 1999, p. 63; Gardner 1984, pp. 211-214). Steinhaus (1999, p. 64) also gives a method for determining how to hit a billiard ball such that it caroms off all four sides before hitting a second ball (Knaster and Steinhaus 1946, Steinhaus 1948).
Click to expand...

This is where I got the idea for a 10 pocket pool table. Its problem of variations.

I am also interested in creating a billiard solver. I like online rts gaming and sometimes AI creates original strategies that work. A billiard solver would be like having a consistent commentary of what the best percentages are.
 
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Patrick Johnson said:
Balls rolling on pool/billiard tables don't rebound at equal angles. Those diagrams show how light would reflect in a room with four mirrored right-angle walls.

pj
chgo
Click to expand...
This. So many other variables in play on a pool table. Cushions, cloth, balls, spin, speed, deflection... the ball will never travel that same absolute path as long as there is human interaction involved and consumables.
 
Pool table physics does not transpire as drawn.

A rolling CB bounces off a rail with (effective) forward spin, arcing it down table. It follows the geometry of reflection only when you consider the reflection to have taken place at the depth from the rails of the diamonds--which is why the diamonds are positioned where they are positioned.

Secondly, your 5-rail shot is unworkable because when a ball bounces off a rail it invariable picks up spin, and bounce 4 consumes that spin removing lots of the energy of the CB.
 
The following link highlights the importance of slipping and rolling.


whats nice is the project has worked through most of the code.

ekiefl.github.io

Creating a billiards simulator: the transition from equations to code

A preliminary implementation of pooltool that supports visualization with pygame
ekiefl.github.io ekiefl.github.io

user is working on more interactions for physics. nice website lots of math, computer code and billiards.

this link gets technical with equations.
ekiefl.github.io

The physics of pool/billiards

A dive into the physics theory behind pool
ekiefl.github.io ekiefl.github.io
 
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2 things to say about billiards and math:

One day at my homeroom I met a player looking to play the local pro. He was ready and anxious to play. The local pro did not show,(he is a "moody" type), so I offered to stand in for cheap since he travelled from across the state for the meetup. He gave me 9-6 and the seven ball. I thought my chances were good until,.......
He ran 4 racks in a row, and I honestly think he missed on purpose to let me on the table. Gotta admit, I was impressed. It was like watching Efren play,.... any angle he went center pocket on the diamond table.
So, I got to talking to him since he was not only dead center in dropping balls, but he was making some really hard angle 2 & 3 cushion shots to evade my safety play. (At this point, I knew I was donating, so I chatted him up on his excellent ability.) He told me he was a Math Professor at a noteable college in the state, So I started inquiry about ball speed and angles. He smiled, and slowly leaked out a few "tidbits" that only a Math major would know. Needless to say: It was the BEST money I ever lost.

The second thing was during my college yrs. I was a business major living off campus. One day I decided to go to the REC Hall as I heard they had a table there. I got there and after signing for the balls, I was handed a white 5 gal. bucket with all the balls. They looked like 1gen aramith that were kept in someones garage for YEARS! I got to the rec hall to see an old Olhausen table that has seen anything BUT a game of pool. There was a coating of dust on the felt which I tried to clean with paper towels and water. There was dirt underneath the felt,.... food spillage on the rails,... (you get the picture).
I could never understand the lack of interest in a game where Math/Physics/Engineering Majors resided. :confused::geek::rolleyes:😖
 
There is actually a small branch of mathematics called "billiards" but it is concerned only with perfect reflections as shown in the diagram above and various shapes that are not always 2:1 rectangles. That branch of math has no relation to physical billiards for the reasons above. Here's an example book you can get for about $50:

1639428700503.png
 
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I remember looking for books like that in the library and immediately putting them back on the shelf :ROFLMAO:

A representative excerpt from the above:


Screen Shot 2021-12-13 at 12.55.29 PM.png
 
Newtons laws of physics fail on a billiard
table.

Specifically the part about space being full of holes.

A ball in motion stays in motion, until it falls into a pocket. What would Newton think happens inside of a space hole?

Newton was good but Riemann was better because he focused on space around objects.

The math around billiards has always fascinated me.
 
skip100 said:
I remember looking for books like that in the library and immediately putting them back on the shelf :ROFLMAO:

A representative excerpt from the above:


View attachment 619751
Click to expand...

Great notation do you want to prove most shots are congruent. Algebra is more powerful and makes programming a dream.

However your notation is still in the early stages of mathematical thoughts
 
MitchAlsup said:
Pool table physics does not transpire as drawn.

A rolling CB bounces off a rail with (effective) forward spin, arcing it down table. It follows the geometry of reflection only when you consider the reflection to have taken place at the depth from the rails of the diamonds--which is why the diamonds are positioned where they are positioned.

Secondly, your 5-rail shot is unworkable because when a ball bounces off a rail it invariable picks up spin, and bounce 4 consumes that spin removing lots of the energy of the CB.
Click to expand...

i have actually made that 5-railer in a match, transposed two diamonds up from the starting position. you go close to the top right corner with left spin, 3rd rail hit will be fairly close to the left side pocket. unless the cloth isn't worn out too much there will be energy enough in the cue ball to land 1.5-2 diamonds from top left corner (and in my case on a dead combo). it's a good shot to know.
 
justnum said:
Newtons laws of physics fail on a billiard table.
Click to expand...
Newtons laws of physics cover everything weighing more than a dram and less than our planet.
Everything that happens on a pool table fits in this domain.

Hint: Newton's laws are perfectly adapted for the energy, velocity, momentum, acceleration, spin, moment of inertia, friction, elastic and inelastic collisions.
 
justnum said:
Newtons laws of physics fail on a billiard table
Click to expand...

Maybe you meant "idealized geometry fails" instead of "Newton's laws of physics fail." "Idealized geometry" definitely fails at a pool table (e.g., when the angles of incidence and reflection are assumed to be equal on kick and bank shots). Newtonian physics explains everything at the pool table extremely well. Many examples, most of which have been validated at the table both anecdotally and through careful experiments, can be found here:

 
Why do I bother reading threads started by certain people? I had sworn off this one but decided to see if anything had changed. It hadn’t.
 
Such great and technical responses.

Thanks

I will get back to figuring out how to make the table of contents to textbooks sound interesting and useful for today's generation or other college bound adults.

When some people are young they find fun things to learn. Some memorize digits in pi and others learn how to play pool.

Pool has been promoted professional and academically. It is fun to revisit that exercise with everyone.

Pool is visually exciting because lining two balls up along two vanishing points makes for some mind blowing perspective changes.

To summarize billiards and math equals "Why you should love and own a billiard table for the Holidays or give as a gift?"

It attracts all kinds of people that use unusual English.
 
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