The number ** π** (pi) is a mathematical constant. It is defined as the ratio of a circle‘s circumference to its diameter, and it also has various equivalent definitions. It appears in many formulas in all areas of mathematics and physics. The earliest known use of the Greek letter

**to represent the ratio of a circle’s circumference to its diameter was by Welsh mathematician William Jones in 1706. It is approximately equal to 3.14159. Pi has been represented by the Greek letter “**

*π**π*” since the mid-18th century, and is spelled out as “

**pi**“. It is also referred to as

**Archimedes’ constant**. [1]

## Continuity: 13, 14, 15

Like all irrational numbers, *π* cannot be represented as a common fraction (also known as a simple or vulgar fraction), by the very definition of irrational number (i.e., not a rational number). But every irrational number, including *π*, can be represented by an infinite series of nested fractions, called a continued fraction. [2]

## Normality: 12

The digits of *π* have no apparent pattern and have passed tests for statistical randomness, including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often. The conjecture that *π* is normal has not been proven or disproven. [3]

## Circular Thinking: 7

Thinking that begins and ends with an assumption (often wrong), also known as circular or paradoxical thinking and in logic called a “logical fallacy.” This process is more common now than in the past, perhaps because of the repetition (“cutting and pasting”) that is found in patient medical records. In circular thinking, a conclusion cannot be proved false or true if it arose from a false premise. Because repeating a statement in circular fashion seems to make it stronger, circular thinking ends by creating statements that sound true and gain wide support . There is no doubt that circular thinking is dangerous and that we must do our best to avoid it. [4]

## Linear Thinking: 19

The opposite of circular thinking is linear (vertical) thinking. In this type of reasoning, progress is made in a step-by-step fashion and a response to each step must exist before advancing to the next one. Although linear thinking advances by logic, it is by its own nature highly focused on single pathways and as such tends to ignore other possibilities and alternatives. Linear thinking is basically a binary process in which answers are “Yes” or “No” (correct or incorrect), excluding all considerations beyond these 2 responses. These features make it fast, organized, and sequential and therefore it is the most common type of thought process used. People generally regard linear thinking as an honest, mature, and intelligent process when in reality it lacks ingenuity, innovation, and originality. Similar to circular thinking, linear thinking is characterized by repetition and is, in the long term, detrimental to intellectual advancement. [5]

## Convergent Thinking: 15

**Convergent thinking** is a term coined by Joy Paul Guilford as the opposite of divergent thinking. It generally means the ability to give the “correct” answer to standard questions that do not require significant creativity, for instance in most tasks in school and on standardized multiple-choice tests for intelligence. Convergent thinking is the type of thinking that focuses on coming up with the single, well-established answer to a problem. It is oriented toward deriving the single best, or most often correct answer to a question. Convergent thinking emphasizes speed, accuracy, and logic and focuses on recognizing the familiar, reapplying techniques, and accumulating stored information. It is most effective in situations where an answer readily exists and simply needs to be either recalled or worked out through decision making strategies. A critical aspect of convergent thinking is that it leads to a single best answer, leaving no room for ambiguity. In this view, answers are either right or wrong. The solution that is derived at the end of the convergent thinking process is the best possible answer the majority of the time. [6]

## Divergent Thinking: 13

Divergent thinking is a thought process or method used to generate creative ideas by exploring many possible solutions. It typically occurs in a spontaneous, free-flowing, “non-linear” manner, such that many ideas are generated in an emergent cognitive fashion. Many possible solutions are explored in a short amount of time, and unexpected connections are drawn. Following divergent thinking, ideas and information are organized and structured using convergent thinking, which follows a particular set of logical steps to arrive at one solution, which in some cases is a “correct” solution. The psychologist J.P. Guilford first coined the terms convergent thinking and divergent thinking in 1956. [7]

## Lateral Thinking: 14

Lateral thinking is a manner of solving problems using an indirect and creative approach via reasoning that is not immediately obvious. It involves ideas that may not be obtainable using only traditional step-by-step logic. Considered pseudo-science by some, the term was first used in 1967 by Edward de Bono in his book *The Use of Lateral Thinking*. De Bono cites the Judgment of Solomon as an example of lateral thinking, where King Solomon resolves a dispute over the parentage of a child by calling for the child to be cut in half, and making his judgment according to the reactions that this order receives. Edward de Bono also links lateral thinking with humour, arguing it entails a switch-over from a familiar pattern to a new, unexpected one. It is this moment of surprise, generating laughter and new insight, which facilitates the ability to see a different thought pattern which initially was not obvious. According to de Bono, lateral thinking deliberately distances itself from the standard perception of creativity as “vertical” logic, the classic method for problem solving. [8]

## Out of Box Thinking: 9

Since the advent of computers, a large number of digits of *π* have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of *π*, and found them consistent with normality; for example, the frequencies of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found. Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the infinite monkey theorem. Thus, because the sequence of *π*‘s digits passes statistical tests for randomness, it contains some sequences of digits that may appear non-random, such as a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of *π*. This is also called the “Feynman point” in mathematical folklore, after Richard Feynman, although no connection to Feynman is known. [9]

The world as we have created it is a process of our thinking.

Albert Einstein

It cannot be changed without changing our thinking.

alter.native“The anomaly showed up most starkly in Base 11 arithmetic, where it could be written out entirely as zeros and ones. Compared with what had been received from Vega, this could be at best a simple message, but its statistical significance was high. The program reassembled the digits into a square raster, an equal number across and down. The first line was an uninterrupted file of zeros, left to right. The second line showed a single numeral one, exactly in the middle, with zeros to the borders, left and right. After a few more lines, an unmistakable arc had formed, composed of ones. The simple geometrical figure had been quickly constructed, line by line, self-reflexive, rich with promise. The last line of the figure emerged, all zeros except for a single centered one. The subsequent line would be zeros only, part of the frame. Hiding in the alternating patterns of digits, deep inside the transcendental number, was a perfect circle, its form traced out by unities in afield of noughts.”

Contact, Carl SAGAN

difference and repetition with return“Is the origin of the procedure that starts with generalities found in abstractions? No! In the field of rhythm, certain very broad concepts nonetheless have specificity: let us immediately cite repetition. No rhythm without repetition in time and in space, without reprises, without returns, in short without measure. But there is no identical absolute repetition. indefinitely. Whence the relation between repetition and difference. When it concerns the everyday, rites, ceremonies, fêtes, rules and laws, there is always something new and unforeseen that introduces itself into the repetitive: difference.”

Rithmanalysis, Henri LEFEBVRE

coincidentia oppositorum“Words sometimes (perhaps always) harbor secrets. A secret is, of course, that which is “kept from knowledge or observation; hidden, concealed.” When used of a place, secret means “retired, remote, lonely, secluded, solitaryfl The word “secret” derives from the Latin secretus (separate, out of the way), which, in turn, comes from the past participle of secernere (to put apart, separate). Joseph Shipley lists three roots for secret: ker, scratch, cut, pluck, gather, dig, separate, sift; sek, cut, scrape, separate, sift; and sue, se, personal relations, one’s own; his, hers, its. “Se, by oneself, came to mean apart, without; in this sense it is an English prefix to innumerable words, as secret, secrete; secure (whence sure); seduce: lead astray; segregate: part from the herd; separate, sever; several: existing apart; sex, cut apart, as Eve from Adam.”‘ For reasons that will become apparent as we proceed, it is not in- significant that script and scripture also deriye from ker and sek. The Danish word for secret is hemmelighed. Hemmelighed carries the trace of hjem, which means “home.” That which is secret is separate, cut off, private; something that is inside or interior, as if restricted to and by the family hearth. There is, however, something strange about hemmelighed. When push- ed too far, its domestic connotations are reversed. Hemmelighedsfuld means mysterious or uncanny. The other Danish word used to designate that which k uncanny is uhyggelig. Hyggelig, an extremely important word in Danish language and culture, has no precise English equivalent. It connotes coziness, comfortableness, and homeyness. Forever returning to disturb the hygglig feeling of the home, u-hyggelig suggests something that can never be domesticated.

The implications of the associations among hjem, hemmelighed, hem- melighedsfuld, and uhyggelig can be clarified by considering their German equivalents. The German word for secret is Geheim, which, like the Danish hemmelighed, is associated with the home: die Heim. Geheim and Heim bear an unexpected relationship to the uncanny—unheimlich. Freud begins his famous essay on “The Uncanny” with a long etymological excursus in which he ex- plores the interplay of unheimlich, Heim, and Geheim:

What interests us most in this tong extract is to find that, among its dif- ferent shades of meaning the word heimlich exhibits, one which is iden- tical with its opposite, unheimlich. . . . In general we are reminded that the word heimlich is not unambiguous, but belongs to two sets of ideas, which without being contradictory are yet very different: on the one hand, it means that which is familiar and congenial, and on the other, that which is concealed and kept out of sight. The word unheimlich is only used customarily, we are told, as the contrary of the first significa- tion, and not of the second. . . On the other hand, we notice that Schelling says something that throws quite a new light on the concept of the “uncanny,” one which we had certainly not awaited. According to him everything is uncanny that ought to have remained hidden and secret, and yet comes to light. Some of the doubts that have thus arisen are removed if we con- sult Grimm’s dictionary. We read:

Heimlub; adj. and adv. vemaculus, orcultus; MIIG. helmelich, hemilîch. . . .4. From the idea of “homelthe,””belonging to the house,” the further idea is developed of something withdrawn from the eyes of others, something concealed, secret, and this idea is expanded in many ways. . . .

Heimlich in a different sense, as withdrawn from knowledge, un- conscious: . . . Heimlich also has the meaning of that which is obscure, inaccessible to knovvledge. . .

9. The notion of something hidden and dangerous, which is expressed in the last paragraph, is still further developed, so that “heimlich” comes to have the meaning usually ascribed to unheimlich.

Thus heimlich is a word the meaning of which develops towards an ambivalence, until it finally coincides with its opposite, unheimlich.”

Tears, Mark C. TAYLOR

to take on the rhythm of a given“We are returning to Kant. May this be an occasion for you to skim, read or re-read The Critique of Pure Reason. There is no doubt that a tremendous event in philosophy happens with this idea of critique. In going into it, ourselves, or in going back into it, I had stopped reading it a very long time ago and I read it again for you, it must be said that it is a completely stifling philosophy. It’s an excessive atmosphere, but if one holds up, and the important thing above all is not to understand, the important thing is to take on the rhythm of a given man, a given writer, a given philosopher, if one holds up, all this northern fog which lands on top of us starts to dissipate, and underneath there is an amazing architecture. When I said to you that a great philosopher is nevertheless someone who invents concepts, in Kant’s case, in this fog, there functions a sort of thinking machine, a sort of creation of concepts that is absolutely frightening. We can try to say that all of the creations and novelties that Kantianism will bring to philosophy turn on a certain problem of time and an entirely new conception of time, a conception of which we can say that its elaboration by Kant will be decisive for all that happened afterwards, which is to say we will try to determine a sort of modern consciousness of time in opposition to a classical or ancient consciousness of time.”

Seminar on Kant, Gilles DELEUZE

Image, Similar and OriginalEvery similar is similar to its original not to each other.

Therefore, all similars are different from each other.

This difference originating from repetition is because repetition repeats itself the same and the different.

The trace of something belongs to its image, not it’s original.

To find the original, it is necessary to stop tracing somewhere.

Law of IdentityIn logic, the law of identity states that each thing is identical to itself. It is the first of the three laws of thought, along with the law of noncontradiction, and the law of excluded middle. However, no system of logic is built on just these laws, and none of these laws provide inference rules, such as modus ponens or DeMorgan’s laws.

In its formal representation, the law of identity is written: “a = a” or “For all x: x = x”, where a or x refer to a term rather than a proposition, and thus the law of identity is not used in propositional logic. It is that which is expressed by the equals sign “=”, the notion of identity or equality. It can also be written less formally as A is A. One statement of such a principle is “A rose is a rose is a rose.”

In logical discourse, violations of the law of identity result in the informal logical fallacy known as equivocation. That is to say, we cannot use the same term in the same discourse while having it signify different senses or meanings without introducing ambiguity into the discourse – even though the different meanings are conventionally prescribed to that term. The law of identity also allows for substitution and is a tautology.

Square Roots of −1Although there are no real square roots of −1, the complex number i satisfies i2 = −1, and as such can be considered as a square root of −1. The only other complex number whose square is −1 is −i because there are exactly two square roots of any non‐zero complex number, which follows from the fundamental theorem of algebra. In the algebra of quaternions – where the fundamental theorem does not apply – which contains the complex numbers, the equation x2 = −1 has infinitely many solutions.

Mathematical BeautyEuler’s identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:

The number 0, the additive identity.

The number 1, the multiplicative identity.

The number π (π = 3.141…), the fundamental circle constant.

The number e (e = 2.718…), a.k.a. Euler’s number, which occurs widely in mathematical analysis.

The number i, the imaginary unit of the complex numbers.