polygonalization

 In computational geometry, a polygonalization of a finite set of points in the Euclidean plane is a simple polygon with the given points as its vertices.

 

Penrose stairs

 The Penrose stairs or Penrose steps, also dubbed the impossible staircase, is an impossible object created by Oscar Reutersvärd in 1937 and later independently discovered and made popular by Lionel Penrose and his son Roger Penrose. A variation on the Penrose triangle, it is a two-dimensional depiction of a staircase in which the stairs make four 90-degree turns as they ascend or descend yet form a continuous loop, so that a person could climb them forever and never get any higher. This is clearly impossible in three-dimensional Euclidean geometry but possible in some non-eucliean geometry like in nil geometry.

 

Milü

 Milü (Chinese: 密率; pinyin: mìlǜ; "close ratio"), also known as Zulü (Zu's ratio), is the name given to an approximation to π (pi) found by Chinese mathematician and astronomer Zu Chongzhi in the 5th century. Using Liu Hui's algorithm (which is based on the areas of regular polygons approximating a circle), Zu famously computed π to be between 3.1415926 and 3.1415927^[1] and gave two rational approximations of π, 22/7 and 355/113, naming them respectively Yuelü (Chinese: 约率; pinyin: yuēlǜ; "approximate ratio") and Milü.

 

vinculum

 A vinculum (from Latin vinculum 'fetter, chain, tie') is a horizontal line used in mathematical notation for various purposes.

 

discretization

 

In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable (creating a dichotomy for modeling purposes, as in binary classification).

Discretization is also related to discrete mathematics, and is an important component of granular computing. In this context, discretization may also refer to modification of variable or category granularity, as when multiple discrete variables are aggregated or multiple discrete categories fused. 

 

quadrature

quadrature (plural quadratures)

1. the process of making something square; squaring
2. (mathematics) [1]  [quotations ▼]
3. (astronomy) a situation in which three celestial bodies form a right-angled triangle, the observer being located at the right angle

   When the Moon is in quadrature, it appears in the sky as a half-moon.

4. (physics) the condition in which the phase angle between two alternating quantities is 90°
5. (art) A painting painted on a wooden panel

quadratrix

quadratrix (plural quadratrices or quadratrixes)

1. (mathematics) A curve having ordinates which are a measure of the area (or quadrature) of another curve.

Tractable

Tractable (meaning "easily managed") may refer to:

1. Capable of being easily led, taught, or managed; docile; manageable; governable. 
2. Capable of being shaped; malleable. 
3. (obsolete) Capable of being handled or touched; palpable; practicable; feasible; serviceable.
4. (mathematics) Sufficiently operationalizable or useful to allow a mathematical calculation to proceed toward a solution. Another involves the use of mathematical closed-form expressions
5. (computer science) Of a decision problem, algorithmically solvable fast enough to be practically relevant, typically in polynomial time.

Piphilology

Piphilology comprises the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant π. The word is a play on the word "pi" itself and of the linguistic field of philology.

undecimal

The undecimal (base-11) positional notation system is based on the number eleven, rather than ten as in decimal or eight in octal and so on. It is not a commonly used system. Undecimal requires eleven symbols representing the decimal numbers 0 through 10. For example, if the symbol for 10 were 'A', the decimal numbers 0 to 24 in undecimal would be: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 20, 21, 22. The undecimal number 1A3 would be 234 in decimal.

Quipus

Quipus (or khipus), sometimes called talking knots, were recording devices historically used in the region of Andean South America. A quipu usually consisted of colored, spun, and plied thread or strings from llama or alpaca hair. It could also be made of cotton cords. The cords contained numeric and other values encoded by knots in a base ten positional system. Quipus might have just a few or up to 2,000 cords.

False precision

False precision (also called overprecision, fake precision, misplaced precision and spurious accuracy) occurs when numerical data are presented in a manner that implies better precision than is actually the case; since precision is a limit to accuracy, this often leads to overconfidence in the accuracy as well.

constructivism

In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its existence, according to constructivism. This viewpoint involves a verificational interpretation of the existence quantifier, which is at odds with its classical interpretation.

There are many forms of constructivism. These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis. Constructivism also includes the study of constructive set theories such as IZF and the study of topos theory.

Pre-Intuitionists

In some circles of mathematical philosophy, the Pre-Intuitionists are considered to be a small but influential group who informally shared similar philosophies on the nature of mathematics. The term itself was used by L. E. J. Brouwer, who in his 1951 lectures at Cambridge described the differences between intuitionism and its predecessors:

Of a totally different orientation [from the "Old Formalist School" of Dedekind, Cantor, Peano, Hilbert, Russell, Zermelo, and Couturat, etc.] was the Pre-Intuitionist School, mainly led by Poincaré, Borel and Lebesgue. These thinkers seem to have maintained a modified observational standpoint for the introduction of natural numbers, for the principle of complete induction [...] For these, even for such theorems as were deduced by means of classical logic, they postulated an existence and exactness independent of language and logic and regarded its non-contradictority as certain, even without logical proof. For the continuum, however, they seem not to have sought an origin strictly extraneous to language and logic.

intuitionism

In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied. Instead, logic and mathematics are the application of internally consistent methods to realize more complex mental constructs.

n-gram vs. engram

Engrams are a hypothetical means by which memory traces are stored as biophysical or biochemical changes in the brain (and other neural tissue) in response to external stimuli.

They are also sometimes thought of as a neural network or fragment of memory, sometimes using a hologram analogy to describe its action in light of results showing that memory appears not to be localized in the brain. The existence of engrams is posited by some scientific theories to explain the persistence of memory and how memories are stored in the brain. The existence of neurologically defined engrams is not significantly disputed, though their exact mechanism and location has been a focus of persistent research for many decades.

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In the fields of computational linguistics and probability, an n-gram is a contiguous sequence of n items from a given sequence of text or speech. The items in question can be phonemes, syllables, letters, words or base pairs according to the application. n-grams are collected from a text or speech corpus.

An n-gram of size 1 is referred to as a "unigram"; size 2 is a "bigram" (or, less commonly, a "digram"); size 3 is a "trigram". Larger sizes are sometimes referred to by the value of n, e.g., "four-gram", "five-gram", and so on.

ignoramus et ignorabimus

The Latin maxim ignoramus et ignorabimus, meaning "we do not know and will not know", stood for a position on the limits of scientific knowledge, in the thought of the nineteenth century. It was given credibility by Emil du Bois-Reymond, a German physiologist, in his Über die Grenzen des Naturerkennens ("On the limits of our understanding of nature") of 1872.

On the 8th of September 1930, the mathematician David Hilbert pronounced his disagreement in a celebrated address to the Society of German Scientists and Physicians, in Königsberg:

We must not believe those, who today, with philosophical bearing and deliberative tone, prophesy the fall of culture and accept the ignorabimus. For us there is no ignorabimus, and in my opinion none whatever in natural science. In opposition to the foolish ignorabimus our slogan shall be: Wir müssen wissen — wir werden wissen! ('We must know — we will know!')

Hilbert worked with other formalists to establish concrete foundations for mathematics in the early 20th century. However, Gödel's incompleteness theorems showed in 1931 that no finite system of axioms, if complex enough to express our usual arithmetic, could ever fulfill the goals of Hilbert's program, demonstrating many of Hilbert's aims impossible, and establishing limits on mathematical knowledge.

sedenion

In abstract algebra, sedenions form a 16-dimensional non-associative algebra over the reals obtained by applying the Cayley–Dickson construction to the octonions. The set of sedenions is denoted by .

The term "sedenion" is also used for other 16-dimensional algebraic structures, such as a tensor product of 2 copies of the quaternions, or the algebra of 4 by 4 matrices over the reals.

cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron, i.e. an Archimedean solid, being vertex-transitive and edge-transitive.

Its dual polyhedron is the rhombic dodecahedron.

syzygy

In mathematics, a syzygy is a relation between the generators of a module M. The set of all such relations is called the "first syzygy module of M". A relation between generators of the first syzygy module is called a "second syzygy" of M, and the set of all such relations is called the "second syzygy module of M". Continuing in this way, we derive the nth syzygy module of M by taking the set of all relations between generators of the (n − 1)^th syzygy module of M. If M is finitely generated over a polynomial ring over a field, this process terminates after a finite number of steps; i.e., eventually there will be no more syzygies (see Hilbert's syzygy theorem). The syzygy modules of M are not unique, for they depend on the choice of generators at each step.