word Metric aka language made un punderstandable

All play and no work mkaes Jill a crazy Jack...In group theory, a word metric on a discrete group {\displaystyle G}G is a way to measure distance between any two elements of {\displaystyle G}G. As the name suggests, the word metric is a metric on {\displaystyle G}G, assigning to any two elements {\displaystyle g}g, {\displaystyle h}h of {\displaystyle G}G a distance {\displaystyle d(g,h)}d(g,h) that measures how efficiently their difference {\displaystyle g^{-1}h}g^{-1}h can be expressed as a word whose letters come from a generating set for the group. The word metric on G is very closely related to the Cayley graph of G: the word metric measures the length of the shortest path in the Cayley graph between two elements of G.

Coiled Alizarine

Curiously deep, the slumber of crimson thoughts:
While breathless, in stodgy viridian
Colorless green ideas sleep furiously.

A generating set for {\displaystyle G}G must first be chosen before a word metric on {\displaystyle G}G is specified. Different choices of a generating set will typically yield different word metrics. While this seems at first to be a weakness in the concept of the word metric, it can be exploited to prove theorems about geometric properties of groups, as is done in geometric group theory.

Examples

The group of integers Z

The group of integers Z is generated by the set {-1,+1}. The integer -3 can be expressed as -1-1-1+1-1, a word of length 5 in these generators. But the word that expresses -3 most efficiently is -1-1-1, a word of length 3. The distance between 0 and -3 in the word metric is therefore equal to 3. More generally, the distance between two integers m and n in the word metric is equal to |m-n|, because the shortest word representing the difference m-n has length equal to |m-n|.

The group {\displaystyle \mathbb {Z} \oplus \mathbb {Z} }\mathbb {Z} \oplus \mathbb {Z} 

For a more illustrative example, the elements of the group {\displaystyle \mathbb {Z} \oplus \mathbb {Z} }\mathbb {Z} \oplus \mathbb {Z} can be thought of as vectors in the Cartesian plane with integer coefficients. The group {\displaystyle \mathbb {Z} \oplus \mathbb {Z} }\mathbb {Z} \oplus \mathbb {Z} is generated by the standard unit vectors {\displaystyle e_{1}=\langle 1,0\rangle }e_{1}=\langle 1,0\rangle , {\displaystyle e_{2}=\langle 0,1\rangle }e_{2}=\langle 0,1\rangle and their inverses {\displaystyle -e_{1}=\langle -1,0\rangle }-e_{1}=\langle -1,0\rangle , {\displaystyle -e_{2}=\langle 0,-1\rangle }-e_{2}=\langle 0,-1\rangle . The Cayley graph of {\displaystyle \mathbb {Z} \oplus \mathbb {Z} }\mathbb {Z} \oplus \mathbb {Z} is the so-called taxicab geometry. It can be pictured in the plane as an infinite square grid of city streets, where each horizontal and vertical line with integer coordinates is a street, and each point of {\displaystyle \mathbb {Z} \oplus \mathbb {Z} }\mathbb {Z} \oplus \mathbb {Z} lies at the intersection of a horizontal and a vertical street. Each horizontal segment between two vertices represents the generating vector {\displaystyle e_{1}}e_{1} or {\displaystyle -e_{1}}-e_{1}, depending on whether the segment is travelled in the forward or backward direction, and each vertical segment represents {\displaystyle e_{2}}e_{2} or {\displaystyle -e_{2}}-e_{2}. A car starting from {\displaystyle \langle 1,2\rangle }\langle 1,2\rangle and travelling along the streets to {\displaystyle \langle -2,4\rangle }\langle -2,4\rangle can make the trip by many different routes. But no matter what route is taken, the car must travel at least |1 - (-2)| = 3 horizontal blocks and at least |2 - 4| = 2 vertical blocks, for a total trip distance of at least 3 + 2 = 5. If the car goes out of its way the trip may be longer, but the minimal distance travelled by the car, equal in value to the word metric between {\displaystyle \langle 1,2\rangle }\langle 1,2\rangle and {\displaystyle \langle -2,4\rangle }\langle -2,4\rangle is therefore equal to 5...Linguists account for unusual nature of this sentence by distinguishing two types of selection: semantic selection (s-selection) and categorical selection (c-selection). Relative to s-selection, the sentence is semantically anomalous — senseless — for three reasons:

• The s-selection of the adjective 'colorless' is violated because it can only describe objects that lack color.
• The s-selection of the adverb 'furiously' is violated because it can only describe activity that is compatible with angry action, and such meanings which are generally incompatible with the activity of sleeping.
• The s-selection of the verb 'sleep' is violated because it can with subject that have the ability to engage in sleep.

However, relative to c-selection, the sentence is structurally well-formed:

• The c-selection of the adverb ‘furiously’ is satisfied, as it combines with the verb ‘sleep’, satisfying the requirement that an adverb modifies a verb.
• The c-selection of the adjectives ‘colorless’ and ‘green’ are satisfied as they combine with noun 'idea', satisfying the requirement that an adjective modifies a noun.
• The c-selection of the intransitive verb ‘sleep’ is satisfied as it combines with the subject 'colorless green ideas’, satisfying the requirement that an intransitive verb combines with a subject.

This leads to the conclusion that although meaningless, the structural integrity of this sentence is high.

In general, given two elements {\displaystyle v=\langle i,j\rangle }v=\langle i,j\rangle and {\displaystyle w=\langle k,l\rangle }w=\langle k,l\rangle of {\displaystyle \mathbb {Z} \oplus \mathbb {Z} }\mathbb {Z} \oplus \mathbb {Z} , the distance between {\displaystyle v}v and {\displaystyle w}w in the word metric is equal to {\displaystyle |i-k|+|j-l|}|i-k|+|j-l|.

Definition

Let G be a group, let S be a generating set for G, and suppose that S is closed under the inverse operation on G. A word over the set S is just a finite sequence {\displaystyle w=s_{1}\ldots s_{L}}w=s_{1}\ldots s_{L} whose entries {\displaystyle s_{1},\ldots ,s_{L}}s_{1},\ldots ,s_{L} are elements of S. The integer L is called the length of the word {\displaystyle w}w. Using the group operation in G, the entries of a word {\displaystyle w=s_{1}\ldots s_{L}}w=s_{1}\ldots s_{L} can be multiplied in order, remembering that the entries are elements of G. The result of this multiplication is an element {\displaystyle {\bar {w}}}{\bar {w}} in the group G, which is called the evaluation of the word w. As a special case, the empty word {\displaystyle w=\emptyset }w=\emptyset has length zero, and its evaluation is the identity element of G.

Given an element g of G, its word norm |g| with respect to the generating set S is defined to be the shortest length of a word {\displaystyle w}w over S whose evaluation {\displaystyle {\bar {w}}}{\bar {w}} is equal to g. Given two elements g,h in G, the distance d(g,h) in the word metric with respect to S is defined to be {\displaystyle |g^{-1}h|}|g^{-1}h|. Equivalently, d(g,h) is the shortest length of a word w over S such that {\displaystyle g{\bar {w}}=h}g{\bar {w}}=h.

The word metric on G satisfies the axioms for a metric, and it is not hard to prove this. The proof of the symmetry axiom d(g,h) = d(h,g) for a metric uses the assumption that the generating set S is closed under inverse.

Variations

The word metric...picture Chomsky in a picture of a picture of Chomsky taking a picture or a computer  or a dream...

Chomsky normal form...Not to be confused with conjunctive normal form.

In formal language theory, a context-free grammar, G, is said to be in Chomsky normal form (first described by Noam Chomsky) if all of its production rules are of the form:[citation needed]

A → BC, or

A → a, or

S → ε,

where A, B, and C are nonterminal symbols, the letter a is a terminal symbol (a symbol that represents a constant value), S is the start symbol, and ε denotes the empty string. Also, neither B nor C may be the start symbol, and the third production rule can only appear if ε is in L(G), the language produced by the context-free grammar G.: 92–93, 106 

Every grammar in Chomsky normal form is context-free, and conversely, every context-free grammar can be transformed into an equivalent one which is in Chomsky normal form and has a size no larger than the square of the original grammar's size...

Nightmare I

Tortured my mind’s eye at its small peephole
sees through the virid glass
the endless ghostly oscillographic stream
Furiously sleep ideas green colorless
Madly awake am I at my small window

Angus McIntosh, 1961
JKrishnamurti mentions the dark little corner of your mind in every talk he ever gave on  the subject of thinking about thinking 

In geometry, a solid angle (symbol: Ω) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle from that point.

In the International System of Units (SI), a solid angle is expressed in a dimensionless unit called a steradian (symbol: sr). One steradian corresponds to one unit of area on the unit sphere surrounding the apex, so an object that blocks all rays from the apex would cover a number of steradians equal to the total surface area of the unit sphere, ${\displaystyle 4\pi }$. Solid angles can also be measured in squares of angular measures such as degrees, minutes, and seconds.

History[edit]

The geometry of the sphere was studied by the Greeks. Euclid's Elements defines:

 the sphere in book XI

 discusses 

various properties of the sphere in book XII 

shows how to inscribe the five regular polyhedra within a sphere in book XIII

 Euclid  includes  a theorem showing that the volume of a sphere varies as the third power of its diameter

Credit for Earlier study and published work due to Eudoxus of Cnidus. 

The volume and area formulas were first determined in Archimedes's On the Sphere and Cylinder by the method of exhaustion....Zenodorus was the first to state that, for a given surface area, the sphere is the solid of maximum volume.^[3]

Archimedes wrote about but did not  provide a formula for dividing a sphere into segments whose volumes are in a given ratio. A solution about  dividing a sphere into segments whose volumes are in a given ratio by means of the parabola and the hyperbola was given by Dionysodorus of Amisus  in the 1st century B.C. some 2000+ years ago now, and a similar framing for the visualization of the same shape manipulation 

 — to construct a segment equal in volume to a given segment

 and in surface to another segment — was solved later by al-Quhi.^[3]

A small object nearby will subtend the same solid angle as a larger object at any distance along the...

In geometry, an angle is subtended by an arc, line segment or any other section of a curve when its two rays pass through the endpoints of that arc, line segment or curve section. Conversely, the arc, line segment or curve section confined within the rays of an angle is regarded as the corresponding subtension of that angle. We also sometimes say that an arc is intercepted or enclosed by that angle.

The precise meaning varies with context. For example, one may speak of the angle subtended by an arc of a circle when the angle's vertex is the centre of the circle.. For example, although the Moon is much smaller than the Sun, it is also much closer to Earth. Indeed, as viewed from any point on Earth, both objects have approximately the same solid angle as well as apparent size. This is evident during a solar eclipse.

Thus one can approximate the solid angle subtended by a small facet having flat surface area dS, orientation ${\displaystyle {\hat {n}}}$, and distance r from the viewer as:

${\displaystyle d\Omega =4\pi \left({\frac {dS}{A}}\right)\,({\hat {r}}\cdot {\hat {n}})}$

where the surface area of a sphere is A = 4πr²...In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.

Equivalently, an inscribed angle is defined by two chords of the circle sharing an endpoint.

The inscribed angle theorem relates the measure of an inscribed angle to that of the central angle subtending the same arc.

The inscribed angle theorem appears as Proposition 20 on Book 3 of Euclid's Elements...

Formulas[edit]

If the intersection points A and B of the legs of the angle with the circle form a diameter, then Θ = 180° is a straight angle. (In radians, Θ = π.)

Let L be the minor arc of the circle between points A and B, and let R be the radius of the circle.^[2]

Central angle. Convex. Is subtended by minor arc L

If the central angle Θ is subtended by L, then

$${\displaystyle 0^{\circ }<\Theta <180^{\circ }\,,\,\,\Theta =\left({\frac {180L}{\pi R}}\right)^{\circ }={\frac {L}{R}}.}$$

PProof (for radians)

The circumference of a circle with radius R is 2πR, and the minor arc L is the (Θ/2π) proportional part of the whole circumference (see arc). So

$${\displaystyle L={\frac {\Theta }{2\pi }}\cdot 2\pi R\,\Rightarrow \,\Theta ={\frac {L}{R}}.}$$

roof (for degrees)

The circumference of a circle with radius R is 2πR, and the minor arc L is the (Θ/360°) proportional part of the whole circumference (see arc). So:

Triangleovoid