User:RDBury/Sandbox

Wolfram Classes

Stephan Wolfram in A New Kind of Science and in several papers dating from the mid-1980's defined four classes into which cellular automata and several other simple computational models can be divided depending on their behavior. While earlier studies in cellular automata tended to try to identify type of patterns for specific rules, Wolfram's classification was the first attempt to classify the rules themselves. In order of complexity the classes are:

• Class 1: Nearly all initial patterns evolve quickly into a stable, homogeneous state. Any randomness in the initial pattern disappears.
• Class 2: Nearly all initial patterns evolve quickly into stable or oscillating structures. Some of the randomness in the initial pattern may filtered out, but some remains. Local changes to the initial pattern tend to remain local.
• Class 3: Nearly all initial patterns evolve in a pseudo-random or chaotic manner. Any stable structures that appear are quickly destroyed by the surrounding noise. Local changes to the initial pattern tend to spread indefinitely.
• Class 4: Nearly all initial patterns evolve into structures that interact in complex and interesting ways. Class 2 type stable or oscillating structures may be the eventual outcome, but the number of steps required to reach this state may be very large, even when the initial pattern is relatively simple. Local changed to the initial pattern may spread indefinitely. Wolfram has conjectured that many, if not all class 4 cellular automata are capable universal computation. This has been proved for Rule 110 and Conway's game of life.

These definitions are qualitative in nature and there is some room for interpretation. According to Wolfram,

...with almost any general classification scheme there are inevitably cases which get assigned to one class by one difition and another class by another definition. And so it is with cellular automata: there are occasionally rules...that show some features of one class and some of another.

Notes:

• ANKoS p231 ff.
• [1]
• [2]
• [3]
• [4]
• [5]

Unicode playground

Polytonic Greek

Upper:	ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ
Lower:	αβγδεζηθικλμνξοπρςστυφχψω
Upper+Acute:	ΆΈΉΊΌΎΏ
Lower+Acute:	άέήίόύώ
Upper+Grave:	ᾺῈῊῚῸῪῺ
Lower+Grave:	ὰὲὴὶὸὺὼ
Upper+Circumflex:
Lower+Circumflex:	ᾶῆῖῦῶ
Upper+Lenis:	ἈἘἨἸὈὨ
Lower+Lenis:	ἀἐἠἰὀὐὠ
Upper+Acute+Lenis:	ἌἜἬἼὌὬ
Lower+Acute+Lenis:	ἄἔἤἴὄὔὤ
Upper+Grave+Lenis:	ἊἚἪἺὊὪ
Lower+Grave+Lenis:	ἂἒἢἲὂὒὢ
Upper+Circumflex+Lenis:	ἎἮἾὮ
Lower+Circumflex+Lenis:	ἆἦἶὖὦ
Upper+Asper:	ἉἙἩἹὉὙὩ
Lower+Asper:	ἁἑἡἱὁὑὡ
Upper+Acute+Asper:	ἍἝἭἽὍὝὭ
Lower+Acute+Asper:	ἅἕἥἵὅὕὥ
Upper+Grave+Asper:	ἋἛἫἻὋὛὫ
Lower+Grave+Asper:	ἃἓἣἳὃὓὣ
Upper+Circumflex+Asper:	ἏἯἿὟὯ
Lower+Circumflex+Asper:	ἇἧἷὗὧ
Upper+Diaeresis:
Lower+Diaeresis:
Upper+Acute+Diaeresis:
Lower+Acute+Diaeresis:
Upper+Grave+Diaeresis:
Lower+Grave+Diaeresis:
Upper+Circumflex+Diaeresis:
Lower+Circumflex+Diaeresis:
Upper+Subscript:	ᾼῌῼ
Lower+Subscript:	ᾳῃῳ
Upper+Acute+Subscript:
Lower+Acute+Subscript:	ᾴῄῴ
Upper+Grave+Subscript:
Lower+Grave+Subscript:	ᾲῂῲ
Upper+Circumflex+Subscript:
Lower+Circumflex+Subscript:	ᾷῇῷ
Upper+Lenis+Subscript:	ᾈᾘᾨ
Lower+Lenis+Subscript:	ᾀᾐᾠ
Upper+Acute+Lenis+Subscript:	ᾌᾜᾬ
Lower+Acute+Lenis+Subscript:	ᾄᾔᾤ
Upper+Grave+Lenis+Subscript:	ᾊᾚᾪ
Lower+Grave+Lenis+Subscript:	ᾂᾒᾢ
Upper+Circumflex+Lenis+Subscript:	ᾎᾞᾮ
Lower+Circumflex+Lenis+Subscript:	ᾆᾖᾦ
Upper+Asper+Subscript:	ᾉᾙᾩ
Lower+Asper+Subscript:	ᾁᾑᾡ
Upper+Acute+Asper+Subscript:	ᾍᾝᾭ
Lower+Acute+Asper+Subscript:	ᾅᾕᾥ
Upper+Grave+Asper+Subscript:	ᾋᾛᾫ
Lower+Grave+Asper+Subscript:	ᾃᾓᾣ
Upper+Circumflex+Asper+Subscript:	ᾏᾟᾯ
Lower+Circumflex+Asper+Subscript:	ᾇᾗᾧ
Upper+Diaeresis+Subscript:
Lower+Diaeresis+Subscript:
Upper+Acute+Diaeresis+Subscript:
Lower+Acute+Diaeresis+Subscript:
Upper+Grave+Diaeresis+Subscript:
Lower+Grave+Diaeresis+Subscript:
Upper+Circumflex+Diaeresis+Subscript:
Lower+Circumflex+Diaeresis+Subscript:

Rules:

Upper only at beginning of word.

Accute, Grave, Circumflex only on vowels, last vowel of syllable

Asper, Lenis only on central vowel of syllables at beginning of word that begin with a vowel (or rho at the beginning or a word or part of double rho).

Diaeresis only on iota and upsilon not at beginning of word.

Subscript only on alpha, eta and omega

Diacritic for diphthong of three vowels appears over central vowel

Apostrophe allowed at end of word

὘὜὚἞἖὎὆἟἗὏὇὞
b ᾰ ᾱ ᾵ Ᾰ Ᾱ ᾽ ι ᾿
c ῀ ῁ ῅ ῍ ῎ ῏
d ῐ ῑ ῒ ΐ ῔ ῕ ῗ Ῐ Ῑ ῜ ῝ ῞ ῟
e ῠ ῡ ΰ ῢ ῤ ῥ ῧ Ῠ Ῡ Ῥ ῭ ΅ `
f ῰ ῱ ῵ ´ ῾ ῿

Compounds

è È é É ê Ê ẽ Ẽ ē Ē e̅ E̅ ĕ Ĕ ė Ė ë Ë ẻ Ẻ e̊ E̊ e̋ E̋ ě Ě e̍ E̍ e̎ E̎ ȅ Ȅ

e̐ E̐ ȇ Ȇ e̒ E̒ e̓ E̓ e̔ E̔ e̕ E̕ e̖ E̖ e̗ E̗ e̘ E̘ e̙ E̙ e̚ E̚ e̛ E̛ e̜ E̜ e̝ E̝ e̞ E̞ e̟ E̟

e̠ E̠ e̡ E̡ e̢ E̢ ẹ Ẹ e̤ E̤ e̥ E̥ e̦ E̦ ȩ Ȩ ę Ę e̩ E̩ e̪ E̪ e̫ E̫ e̬ E̬ ḙ Ḙ e̮ E̮ e̯ E̯

ḛ Ḛ e̱ E̱ e̲ E̲ e̳ E̳ e̴ E̴ e̵ E̵ e̶ E̶ e̷ E̷ e̸ E̸ e̹ E̹ e̺ E̺ e̻ E̻ e̼ E̼ e̽ E̽ e̾ E̾ e̿ E̿

0̇ 1̇ 2̇ 3̇ 4̇ 5̇ 6̇ 7̇ 8̇ 9̇ ṅ 0̇ 1̇ 2̇ 3̇ 4̇ 5̇ 6̇ 7̇ 8̇

0, 1, 2, 3, 4, 5, 6, 7, 8 ${\dot {0}},{\dot {1}},{\dot {2}},{\dot {3}},{\dot {4}},{\dot {5}},{\dot {6}},{\dot {7}},{\dot {8}}$ $\textstyle {\dot {0}},{\dot {1}},{\dot {2}},{\dot {3}},{\dot {4}},{\dot {5}},{\dot {6}},{\dot {7}},{\dot {8}}$ $\scriptstyle {\dot {0}},{\dot {1}},{\dot {2}},{\dot {3}},{\dot {4}},{\dot {5}},{\dot {6}},{\dot {7}},{\dot {8}}$

ĖėĠġİıŻż ė Ė ë Ë

Table Playground

Mathematics Abscissa Algebra Algebraic Forms Aliquot Amicable Numbers Angle Argument Arithmetic Axis Bessel Function Binomial Biquadratic Bisectrix Cardioid Catenary Circle Cissoid Cylinder

Determinant Diagonal Diameter Differences, Calculus of Differential Equation Dimension Dodecahedron Ellipse Ellipsoid Epicycloid Equation Figurate Numbers Focus Folium Fourier's Series Frustum Function Geometrical Continuity Geometry

Gnomon Graphical Methods Groups, Theory of Harmonic Harmonic Analysis Hyperbola Icosahedron Infinite Infinitesimal Calculus Interpolation Inversion Involution Lemniscate Limaçon Line Locus Logarithm Logocyclic Curve, Strophoid or Foliate

Magic Square Maxima and Minima Mensuration Number Number, Partition of Numeral Octahedron Ordinate Oval Parabola Perspective Polygon Polygonal Numbers Polyhedral Numbers Polyhedron Porism Prism Probability Projection

Quadradrix Quaternions Roulette Series Serpentine Sphere Spherical Harmonics Spheroid Spiral Surface Table, Mathematical Tetrahedron Triangle Trigonometry Trisectrix Variations, Calculus Vector Analysis Witch of Agnesi Zero

Image Playground

Rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble
Greek rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble rubble.

BLP

Some additional information, the Biographies guidelines state:

Exert great care in using material from primary sources. Do not use, for example, public records that include personal details--such as date of birth, home value, traffic citations, vehicle registrations, and home or business addresses--or trial transcripts and other court records or public documents, unless a reliable secondary source has already cited them.

Presumably this would include searching FEC records or using the Donor Lookup function on OpenSecrets.Org.

Meander curve

A Meander curve is one of a family of curves used to model several phenomena including river meanders.

Bibliography

• http://eps.berkeley.edu/people/lunaleopold/(095)%20River%20Meanders%20-%20Theory%20of%20Minimum%20Variance.pdf

River Meanders--theory of Minimum Variance by W.B. Langbein, L.B. Leopold

Circle

In Euclidean geometry, a circle is that set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the centre. Circles are simple closed curves, dividing the plane into an interior and exterior. Sometimes the word circle is used to mean the interior, with the circle itself called the circumference, which more usually means the length of the circle.

In coordinate geometry a circle with centre (x0,y0) and radius r is the set of all points (x,y) such that

(x - x0)2 + (y - y0)2 = r2

A circle is thus a kind of conic section, with eccentricity zero. All circles are similar, so the ratio between the circumference and radius and that between the area and radius square are both constants. These are 2π and π, respectively, and this is the best known definitions of that constant.

A line cutting a circle in two places is called a secant, and a line touching the circle in one place is called a tangent. The tangent lines are necessarily perpendicular to the radii, segments connecting the centre to a point on the circle, whose length matches the definition given above. The segment of a secant bound by the circle is called a chord, and the longest chord is that which passes through the centre, called the diameter and divided into two radii.

A segment of a circle bound by two radii is called an arc, and the ratio between the length of an arc and the radius defines the angle between them in radians. Some theorems should be mentioned here.

In affine geometry all circles and ellipses become congruent, and in projective geometry the other conic sections join them. In topology all simple closed curves are homeomorphic to circles, and the word circle is often applied to them as a result. The 3-dimensional analog of the circle is the sphere.

Length of the circle's circumference = 2 × pi × radius

Area of the circle = pi × square(radius)

Circles are simple shapes of Euclidean geometry. It is the locus of all points in a plane at a constant distance, called the radius, from a fixed point, called the center. Through any three points not on the same line, there passes one and only one circle.

A chord of a circle is a line segment whose both endpoints lie on the circle. A diameter is a chord passing through the center. The length of a diameter is twice the radius. A diameter is the largest chord in a circle.

Circles are simple closed curves which divide the plane into an interior and an exterior. The circumference of a circle is the perimeter of the circle, and the interior of the circle is called a disk. An arc is any connected part of a circle.

A circle is a special ellipse in which the two foci are coincident. Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.

A circle in the geometry is a round two-dimensional figure that is formed by all points that same distance to a chosen point. The choice is the focal point, indicated m in the figure, and the chosen distance is called the jet, with r indicated in the figure.

Sometimes to the size of a circle to indicate the radius instead used the diameter (d in the figure). This is the greatest distance between two points of the circle, and exactly 2 times as large as the jet.

Sometimes the circle is not the curve on the outside, but the collection of all points within that curve. Mathematically speaking, that is incorrect, all points within a circle forms a disk.

A segment on the border on the circle, called a chord. Each chord passing through the centre of the circle has a diameter of that circle. The length of the diameter is the diameter.

In Euclidean geometry, a circle is the place of the points of the plan that are situated at a distance date, said radius from the circle, from a fixed point, said the centre circle. The circles are simple closed curves, which divide the floor in an interior and exterior. They are conical with eccentricity nothing. The plan contained in a circle along the circumference, is called circle.

The term district is one of the most important terms of plane geometry. A circle is defined as the quantity (geometric place) of all points of the Euclidean levels, the distance from a given point M equal to a fixed number of positive r is rational. The circle is also the location of all points line with this property.

The term circle has several meanings derived from its original meaning geometric. In its first sense, the circle is "round", the ideal figure which reduces the form of numerous natural or artificial objects: the Sun, an eye, the circumference of a tree or a wheel.

For a long time, the current language employed the term both to appoint the curve (circumference) that it delineates the surface. Nowadays, mathematics, the circle is limited to the curve, the surface is called disk.

A circle is a figure without any angle. A circle is defined by a set of points at equal distance from a known center of the circle.