In common mathematical parlance, a mathematical result is called **folklore** if it is an unpublished result with no clear originator, but which is well-circulated and believed to be true among the specialists. More specifically, **folk mathematics**, or **mathematical folklore**, is the body of theorems, definitions, proofs, facts or techniques that circulate among mathematicians by word of mouth, but have not yet appeared in print, either in books or in scholarly journals.

Quite important at times for researchers are **folk theorems**, which are results known, at least to experts in a field, and are considered to have established status, though not published in complete form. Sometimes, these are only alluded to in the public literature.
An example is a book of exercises, described on the back cover:

This book contains almost 350 exercises in the basics of ring theory. The problems form the "folklore" of ring theory, and the solutions are given in as much detail as possible.

Another distinct category is **well-knowable** mathematics, a term introduced by John Conway. These mathematical matters are known and factual, but not in active circulation in relation with current research (i.e., untrendy). Both of these concepts are attempts to describe the actual context in which research work is done.

Some people, in particular non-mathematicians, use the term *folk mathematics* to refer to the informal mathematics studied in many ethno-cultural studies of mathematics. Although the term "mathematical folklore" can also be used within the mathematics circle to describe the various aspects of their esoteric culture and practices (e.g., slang, proverb, limerick, joke).

When researchers in mathematics write grant requests, they are compelled to justify their work to taxpayers and speculate on possible future applications of their research.In the traditional dance of researchers and grant givers, this is well understood: the mathematicians know they are allowed some wishful leeway, and the reviewers generally expect and accept that. Besides, math research is dirt cheap.

But when we talk to the general public, I feel that this overselling is damaging. It misrepresents many of the reasons behind the mathematical endeavor: curiosity, a search of beauty, and a sense of awe which are wholly decoupled from applications…

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Feferman on Foundations(about which more in a follow-up post), we learn that shortly before his death Sol Feferman proposed to OUP a sequel to his terrific volume of papersIn the Light of Logic. He wanted to collect together some more of his later papers of broader philosophical interest, under the suggested titleLogic, Mathematics, and Conceptual Structuralism.

In mathematics and physics, a **brachistochrone curve** (from Ancient Greek * *βράχιστος χρόνος* (brákhistos khrónos)* 'shortest time'), or curve of fastest descent, is the one lying on the plane between a point *A* and a lower point *B*, where *B* is not directly below *A*, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time. The problem was posed by Johann Bernoulli in 1696.

The brachistochrone curve is the same shape as the tautochrone curve; both are cycloids. However, the portion of the cycloid used for each of the two varies. More specifically, the brachistochrone can use up to a complete rotation of the cycloid (at the limit when A and B are at the same level), but always starts at a cusp. In contrast, the tautochrone problem can only use up to the first half rotation, and always ends at the horizontal. The problem can be solved using tools from the calculus of variations and optimal control.

The curve is independent of both the mass of the test body and the local strength of gravity. Only a parameter is chosen so that the curve fits the starting point *A* and the ending point *B*. If the body is given an initial velocity at *A*, or if friction is taken into account, then the curve that minimizes time will differ from the tautochrone curve.

There are many circumstances in which uncertainty is warranted.

^{1}Gas temperature measurements, weather forecasts, horse races, coin flips, and clinical trials all have some uncertainty involved. Probability theory is the science that finds commonality among these seemingly disconnected phenomena. We can observe, for example, that the summation of many “repeatable” random events, properly normalized, begins to look like a gaussian distribution (aka the central limit theorem). We can notice common shapes in the histograms of these repeatable experiments, such as “fat-tailed” or “power law” distributions. And if the event is not repeatable, we can at least apply the rules of probability theory to avoid inconsistencies in our thinking (which would allow a savvy adversary to take advantage of us when gambling).However, I believe there are tasteful and distasteful applications of probability theory. This is because the application of probability to a particular event requires a suspension of disbelief. To consider an event as repeatable and iid is to accept that the causal factors driving the outcome are (practically) unobservable and therefore ignorable. In effect, it means giving up on deeply understanding a causal explanation of the phenomena and instead sweeping the details under the rug of “the distribution.”

This makes probability theory the science of last resort. Only after truly exhausting your ability to investigate causal factors and processes should you indulge in probabilistic thinking. Doing otherwise is a cop-out, one that dangerously

feels“scientific.”

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A fundamental relation in celestial mechanics is Kepler’s equation, linking an orbit’s mean anomaly to its eccentric anomaly and eccentricity. Being transcendental, the equation cannot be directly solved for eccentric anomaly by conventional treatments; much work has been devoted to approximate methods. Here, we give an explicit integral solution, utilizing methods recently applied to the ‘geometric goat problem’ and to the dynamics of spherical collapse. The solution is given as a ratio of contour integrals; these can be efficiently computed via numerical integration for arbitrary eccentricities. The method is found to be highly accurate in practice, with our C++ implementation outperforming conventional root-finding and series approaches by a factor greater than two.

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This post is about a neat trick that is certainly not of my invention, but that should really be more well-known; at least, I haven’t heard of it till I stumbled across it while reading Box2D sources.

There are a lot of bounding volumes out there; the most widespread are certainly spheres and boxes, which come in two flavors - axis-aligned bounding boxes (AABB) with faces parallel to the coordinate planes, and oriented bounding boxes (OBB), which is essentially a AABB and an orientation matrix.

It’s common to use AABB in spatial subdivision structures, like octrees, kD-trees, ABT and so on - the intersection test between two AABB is pretty straightforward. However, when dealing with dynamic meshes, it is needed to recalculate the AABB of the mesh when the mesh transformation changes.

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I’ve been curious about Gottfried Leibniz for years, not least because he seems to have wanted to build something like

Mathematicaand Wolfram|Alpha, and perhapsA New Kind of Scienceas well—though three centuries too early. So when I took a trip recently to Germany, I was excited to be able to visit his archive in Hanover.Leafing through his yellowed (but still robust enough for me to touch) pages of notes, I felt a certain connection—as I tried to imagine what he was thinking when he wrote them, and tried to relate what I saw in them to what we now know after three more centuries:

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How many people would you need at a party to guarantee that at least three individuals know each other, or that at least three do not know each other? Working this out with pen and paper may take a while, but many mathematicians will readily tell you the answer is six. This party scenario, also called the “friends and strangers” theorem, is based on a concept known as Ramsey numbers, named after early 20th-century British Mathematician Frank Ramsey.

Now, imagine that more people are invited to this hypothetical party. How many would be required to ensure that at least five people know each other, or, conversely, that at least five are strangers? The answer is not clear. Indeed, mathematicians only know that the number of required people would be at least 43 and no more than 48. The actual answer falls somewhere in this range but is unknown. Add even more people to the party, and the uncertainty in the problem quickly becomes enormous.

Now, for the first time in decades, Caltech professor of mathematics David Conlon and his colleague Asaf Ferber of UC Irvine have shrunk that uncertainty by exponential amounts, for a special category of Ramsey numbers known as multicolor Ramsey numbers. The researchers describe their work in a study appearing in the journal

Advances in Mathematics.

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