In my recent post on mapping Perlin noise to angles, I was put on to the subject of Curl noise, which I thought I understood, but did not. I figured out what Curl noise

reallywas in a subsequent post and then posted my earlier incorrect (but still interesting and perhaps useful) concept of Curl noise in yet another post. Although I kind of understood what Curl noise was at that point, I wanted to give myself a more complete understanding, which I usually do by digging into the code, making sure I understand every line 100% and seeing what else I can do with it, trying to make multiple visualizations with it to test my understanding, etc.

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Mathematician: That depends on what you mean by “universe.” Here’s a framing:A circle of radius R centered at a point P is the set of all points in the plane with distance R from P. The diameter D of this circle is twice the radius, but can also be thought of as the longest possible straight-line path from a point on the circle to another point on the circle. The circumference C of the circle is its arc length. By definition, Pi = C/D.

The Collatz conjecture is an allegedly-unsolved problem in mathematics.

I say allegedly there because, reading those words, I feel like mathematicians are going to jump out from my closet yelling, “You got punk’d!” for being duped into believing it’s unsolved.

We’ve all heard of integers, rationals, reals, even complex numbers, but what on earth are surreal numbers? They are a beautiful way of defining a class of numbers which includes all reals, but also ordinal numbers; i.e. all the different infinities and even infinitesimal numbers. Not only this but we get a full system of arithmetic for all these numbers. Ever wondered what (∞−1) is, or √∞ ? Before we get stuck into that, let’s learn some history.

It all started a long, long time ago in a galaxy far, far away (Cambridge in the 1970s). A man by the name of John H. Conway was playing Go, an ancient Chinese game that’s very elegant in itself. After much thought, he realised that the later stages of the game could be thought of as the sum of many smaller games. Conway then applied his ideas to other games like Checkers and Dominoes. It seemed that these games were behaving as if they were numbers.

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**Squaring the circle** is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.

In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi (π) is a transcendental, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if π were transcendental, but π was not proven transcendental until 1882. Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a finite number of steps, since there are rational numbers arbitrarily close to π.

The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.

The term *quadrature of the circle* is sometimes used to mean the same thing as squaring the circle, but it may also refer to approximate or numerical methods for finding the area of a circle.

The

Tower of Hanoiis a famous mathematical puzzle. A set of disks of different sizes are stacked like a cone on one of three rods, and the challenge is to move them onto another rod while respecting strict constraints:

Only one disk can be moved at a time.

No disk can be placed upon a smaller one.

Many years ago, I read in Ripley’s

Believe It or Notcolumn that, in a Buddhist temple in India, there is a large room with three stout pillars on which 64 golden disks are stacked. A team of monks work tirelessly, transferring these disks between the three pillars. With one movement each second, the time required to solve the puzzle is about 585 billion years, at which time the world will either end, or enter another cycle.However, Ripley was not always a reliable source: in fact, the puzzle was invented in 1883 by the French mathematician Édouard Lucas. The minimum number of moves required to solve the puzzle with disks is , the -th Mersenne number. There are algorithms to ensure a solution in the minimum number of moves (see Wikipedia page referenced below).

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The “essence of mathematics”, which we have tried to capture in these problems is mostly implicit, and so is often left for the reader to extract. Occasionally it has seemed appropriate to underline some aspect of a particular problem or its solution. Some comments of this kind have been included in the text that is interspersed between the problems. But in many instances, the comment or observation that needs to be made can only be appreciated after readers have struggled to solve a problem for themselves. In such cases, positioning the observation in the main text might risk spilling the beans prematurely. Hence, many important observations are buried away in the solutions, or in the Notes which follow many of the solutions. More often still, we have chosen to make no explicit remark, but have simply tried to shape and to group the problems in such a way that the intended message is conveyed silently by the problems themselves.

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(application/pdf - 5.81 MB)

**Reverse mathematics** is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones.

The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory.

Reverse mathematics is usually carried out using subsystems of second-order arithmetic, where many of its definitions and methods are inspired by previous work in constructive analysis and proof theory. The use of second-order arithmetic also allows many techniques from recursion theory to be employed; many results in reverse mathematics have corresponding results in computable analysis. Recently, *higher-order* reverse mathematics has been introduced, in which the focus is on subsystems of higher-order arithmetic, and the associated richer language.

The program was founded by Harvey Friedman (1975, 1976) and brought forward by Steve Simpson. A standard reference for the subject is (Simpson 2009), while an introduction for non-specialists is (Stillwell 2018). An introduction to higher-order reverse mathematics, and also the founding paper, is (Kohlenbach (2005)).

In preparation for a conference entitled “Distributed Consensus with Cellular Automata and Related Systems” that we’re organizing withNKN(= “New Kind of Network”) I decided to explore the problem of distributed consensus using methods fromA New Kind of Science(yes, NKN “rhymes” with NKS) as well as from theWolfram Physics Project.

Multiplying huge integers of

`n`

digits can be done in time`O(nlog(n))`

using Fast Fourier Transforms (FFT), instead of the`O(n2)`

time complexity normally required. In this paper we present this technique from the view-point of polynomial multiplication, explaining a recursive divide-and-conquer approach to FFT multiplication. We have compared both methods of multiplication quantitatively and present our results and conclusions in this paper, along with complexity analyses and error bounds.

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The **15 theorem** or **Conway–Schneeberger Fifteen Theorem**, proved by John H. Conway and W. A. Schneeberger in 1993, states that if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers. The proof was complicated, and was never published. Manjul Bhargava found a much simpler proof which was published in 2000.

Bhargava used the occasion of his receiving the 2005 SASTRA Ramanujan Prize to announce that he and Jonathan P. Hanke had cracked Conway's conjecture that a similar theorem holds for integral quadratic forms, with the constant 15 replaced by 290. The proof has since appeared in preprint form.

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).