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This is an incredibly naive question so this may be closed. Nevertheless, I have been reading about the problem asking if every obtuse triangle admits a periodic billiard path, which has been open for a very long time. As someone who has not worked on this problem, I am wondering why what (on the surface) appears to be a "simple" problem is in fact so difficult to solve.

From the little I have read, it would appear that there has indeed been progress into the problem by the likes of Schwartz, Halbeisen et al., Vorobets et al., and more, however none have actually solved this problem. I find it curious that finding periodic billiard paths for acute triangles via the Fagnano billiard orbits is so natural and even simple, yet as soon as the same question is asked about right or obtuse triangles the ease in answering the question is vanquished.

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Start from the simplest path, a triangle with angles $\alpha, \beta, \gamma$, and build the unique triangle for which this path is a billiard path. It's easy to see that the latter triangle has angles $\frac{\alpha+\beta}{2}, \frac{\gamma+\beta}{2}, \frac{\alpha+\gamma}{2}$ and is therefore acute. Any acute triangle can be obtained in such a way.

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In any case, before we can comment on the smoothness of the Earth compared with a billiard ball, I think we require more information on either WPA rules or manufacturing standards. [Edit: Thanks to commenter Mark Folsom for providing the following clarification of just how smooth a billiard ball is:

First, a minor nit: neither of the quoted diameters is correct to the given number of significant digits. But that will not affect our calculations here. What the article seems to miss is that the stated tolerance of a billiard ball diameter is plus or minus 0.005 inch. That is, the diameter may be as small as 2.245 inches, or as large as 2.255 inches. Enlarging this 0.01 inch difference to the scale of the Earth, the allowable difference in diameters is about 56.6 km, more than the actual difference of 42.8 km. So the Earth is indeed as round as a regulation billiard ball.

Having said all this, I think this entire analysis abuses the spirit of the law, so to speak. The WPA probably does not intend to allow such ellipsoidal billiard balls onto pool tables around the world, but rather to allow some variability in the size of nearly-spherical balls. That is, the intent of the regulation is more likely that a ball should be spherical with a fixed diameter, but that diameter may be 2.245 inches for one ball, and 2.255 inches for another ball.

Has anyone actually measured the diameter differences on a billiard ball? I guess the quality manufactures make them as round as possible and with better smothness and tolerances than the extreme tolerances allow.

The question should be : Is earth smoother than a billiard ball with the worst tolerances?

We seem to be in violent agreement. I agree, as also indicated in the last paragraph of the post, that nominal diameter tolerance (or something like it) is probably the more likely intended meaning of the WPA requirement. But what, *if any*, is the corresponding sphericity requirement? If it is in fact G1000, then you are also correct that the Earth is not as *spherical* as a billiard ball. But I think we need more information, preferably from an actual billiard ball manufacturer, before simply assuming that billiard ball manufacturing (where resin is the typical material being manipulated) borrows all equipment, specifications, etc., from (steel) ball bearing manufacturing.

This is the additional information we are looking for. In comparison, at the scale of a billiard ball, the Mariana Trench is a groove almost 2000 microinches deep. So it seems the Earth is nowhere near as smooth as a billiard ball.

id say that if you consider the surface of the earth to include the water surfaces, then the rms roughness is likely to drop to a similar level as the billiard ball, statistically anyway. The main issue is the lack of info on sphericity as has been stated.

Quite amazing really, when you look at a cliff. Also amazing if you look at a billiard ball and imagine that those invisible scratches are the height of Everest and how flipping tiny we are in comparison.

We need to hear from Skip D the OP, who asked for a billiard table, not a pool table. Billiard tables have no pockets. One or more of those shown at the 3D Warehouse site are true billiards tables, but the great majority have pockets.

I've a friend who has an expensive billiard table and plays. Its slate top has electric heating elements to regulate its temperature to a tournament-level standard. He has a smartphone app to turn it on from wherever he is so it is ready to play when he gets there.

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There are also variations that are like subcategories. If you put a saddle stem on a billiard, you have a Saddle Billiard. You can also have an oval shank or a diamond shank or a thin "pencil" shank. The bowl and/or shank can have flat panels resulting in a Panel Billiard. A box-shaped bowl with a square shank is a "Four-Square" Billiard.

The Women\u2019s Professional Billiard Association (WPBA) is the governing body of women\u2019s professional billiards in the U.S.A., and one of the longest-running women's professional sports organizations in the world.

At the time, billiards was at the peak of its popularity. However, as the game found more and more enthusiasts, a problem arose. Strangely enough, the problem was linked to a shortage of elephants, which were already scarce at that time! For over two centuries, ivory from elephant tusks had been the only material both used and usable to fashion billiard balls, and there was no longer enough ivory to meet increasing demand, especially given that a single tusk only produces 4 to 5 balls!

The Civil War and the blockade imposed on the southern states made it impossible to import the highly sought-after material and ultimately put an end to the trade.

Seeing its industry threatened, the Phella and Collender company, a manufacturer of billiard accessories, came up with the idea of launching a competition offering a reward to anyone who could come up with a substitute for this now banned material. John Wesley Hyatt, a young inventor, started his research using cellulose nitrate. However, tragedy struck and he cut his finger. While patching himself up, he noticed that the collodion - an antiseptic of the time - which he accidentally spilled on his preparation, made the whole thing harden.

Convinced of having stumbled upon the discovery of the century, he continued his research. It would take him another seven years to find the solution by adding camphor. A patent was filed on 12 July 1870: Celluloid was born and would pave the way for what would later become known as plastics.

Celluloid was used to manufacture billiard balls for some time, but it was too fragile to resist the impact of the balls against each other and was ultimately replaced by other synthetic materials, such as Bakelite, in the 20th century. Nowadays, billiard balls are mainly made of phenolic resin, a material that behaves similarly to ivory, and have a few additional qualities: they get dirty less quickly, they are perfectly spherical and they last longer.

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The cloth must be non-directional, nap-free billiard fabric which will not pill or fluff, composed of composed of between 80% and 85% combed worsted wool, and between 15% to 20% nylon. 100% combed worsted wool fabric is preferred. No backed cloth will be allowed. Only the colors of yellow-green, blue-green or electric blue are acceptable for WPA competition.

The WPA recommends only the colors green and blue for chalk. A soft (horse hair) brush, a cloth-cleaner made with billiard fabric, or a brushless (without rotating brush) vacuum cleaner are the recommended table and cushion cleaning devices. Brushes that shed bristles are not recommended.

The mechanical bridge, also called rake, crutch or rest, is an accessory of the billiard sports table and consists of a stick with a bridge head mounted at its end to support the shaft of the cue stick replacing the hand bridge during shots difficult to reach. The stick or handle of the mechanical bridge is very similar in shape to the cue stick. The bridge

head has notches or grooves, usually at various heights, in which the cue shaft can rest. The contour of the bridge head should be smooth in order not to mar the cue shaft or rip the threads of the table-cloth when being used. 2351a5e196