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

Math Symbols

∈∉⊆∪∩∅←→≤≥≠⋅

Table Playground

Differences, Calculus of

Differential Equation

Geometrical Continuity

Infinitesimal Calculus

Logocyclic Curve, Strophoid or Foliate

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.

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.

PTA notes

RPG

A role-playing video game (RPG) is one of a loosely defined genre of computer and video games with origins in pencil and paper role-playing games such as Dungeons & Dragons, borrowing much of their terminology, settings and game mechanics.

Characteristics of the genre

While no single feature or characteristic of a video game can be used to identify it as an RPG, there are several characteristics of the genre as a whole.

Players usually control a small number of characters called a party. This is in contrast to the wargame genre where players control large groups of identical units, as well as non-humanoid units such as tanks and airplanes.^[1] This is also in contrast to adventure games where the player controls a single character.
Characters gradually grow in power and abilities, usually by defeating various hostile creatures through the mechanism of experience points. These can be extremely in-depth and complicated and many games give the player some degree of latitude in determining which abilities will be improved. Most RPGs, in common with paper based games, use level as a broad measure of the character's strength.
The game world is inhabited by many computer controlled characters with which the player's character can interact. These

non-player characters (NPCs) run shops with equipment and supplies, ask the player to complete quests in return for a reward, give advice and information about playing the game, or simply provide additional color.

There is a highly developed story which unfolds as the game progresses.
The game world is large and diverse. In some cases the exploration of new areas is limited by the ability of characters to survive attacks by their inhabitants. In others, the player must reach a specific point in the story before a new area is accessible.
The game depends very little the player's physical skills such as timing and aim^[1] . Often the player controls the characters actions by issuing commands though a system of menus. The effectiveness of these actions determined by the character's numeric attributes.
The game depends correspondingly more on the player's strategy, tactics and knowledge of the game. For example, the outcome of a battle may depend on players exploiting a weakness to a specific type of weapon.
In the past, most RPGs entered a separate "battle mode" when enemies were encountered. Recently there is a trend toward incorporating battles seamlessly into the rest of the game. This can be seen in the Final Fantasy series where Final Fantasy X and earlier have a battle mode while Final Fantasy XI and later use a seamless approach.
The game often has a fantasy setting with primitive technology enhanced by magic.
The games usually feature a rich collection of items to collect and manage. Success in the game is strongly affected by whether characters have the proper armor, weapons and other equipment.
The different members of the party play different roles in battle according to their specific abilities and weaknesses. For example a party will usually have a character that specializes in doing physical damage to enemies and another that specializes in healing other members of the party.

Euclid's Elements

Book 1 Definitions 1: Point 2: Line 3: Extremities of lines 4: Straight line 5: Surface 6: Extremities of Surfaces 7: Plane surface 8: Plane angle 9: Rectilineal angle 10: Right angles 11: Obtuse angle 12: Acute angle 13: Boundary 14: Figure 15: Circle 16: Circle center 17: Circle diameter 18: Semicircle 19: Rectilineal figure 20: Equilateral, isosceles, scalene triangles 21: Right, obtuse, acute angled triangles 22: Square, oblong, rhombus, rhomboid, trapezia 23: Parallel lines Postulates 1: Draw a straight line on two points 2: Produce a finite straight line 3: Draw a circle with given center and distance 4: All right angles are equal 5: Intersection of two straight lines on a third straight lines so the included angles are less than two right angles Common Notions 1: Things equal to the same thing are equal 2: Equals added to equals are equal 3: Equals subtracted from equals are equal 4: Things which coincide are equal 5: A whole is greater than a part Propositions 1: Construct an equilateral triangle 2: Mark a segment on a given straight line equal to a given straight line segment 3: Cut from a straight line segment a segment equal to a given shorter line segment 4: Side-Angle-Side 5: Isosceles triangle theorem (Pons asinurum) 6: Isosceles triangle theorem converse. 7: Length of sides of a triangle on a given base determine the triangle. 8: Side-Side-Side. 9: Construct an angle bisector. 10: Construct a bisector of a line segment. 11: Construct a line perpendicular to a given line at a given point on the line. 12: Construct a line perpendicular to a given line through given point not on the line. 13: Adjacent angles on a line equal two right angles. 14: Converse to 13. 15: Opposite angles 16: Exterior angle theorem 17: Two angles in a triangle are less than two right angles. 18: Greater side subtends greater angle 19: Greater angle subtended by greater side 20: Two sides of a triangle greater than the remaining side (triangle inequality) 21: Triangle within another triangle on the same base has smaller sides and greater angle. 22: Construct triangle with sides equal to given segments. 23: On a given line, construct an angle equal to a given angle at a given point. 24: Given two triangles with two equal sides, the triangle with the greater angle will have the greater base. 25: Converse of 24 26: Angle-Side-Angle, Side-Angle-Angle 27: If alternate angles are equal then the lines are parallel 28: In interior angles equal two right angles then the lines are parallel 29: Converse to 27 and 28 30: Lines parallel to the same line are parallel 31: Construct a line parallel to a given line through a given point. 32: Angles in a triangle are two right angles. 33: Lines joining equal and parallel segments are equal and parallel 34: In a parallelogram, opposite sides and angles are equal, a diameter bisects the areas. 35: Parallelograms having the same base and equal parallels are equal 36: Parallelograms having the equal bases and the same parallels are equal 37: Triangles on the same base and in the same parallels are equal 38: Triangles on equal bases and in the same parallels are equal 39: Equal triangles on the same base are in the same parallels 40: Equal triangles on equal bases are in the same parallels 41: Parallelogram is double a triangle on the same base and the same parallels 42: Construct a parallelogram in a given angle equal to a given triangle. 43: In a parallelogram, the complements of parallelogram about a diameter are equal. 44: Construct a parallelogram on a given line equal to a given triangle. 45: Construct a parallelogram in a given angle equal to a given figure 46: Construct a square on a given line. 47: Pythagorean theorem 48: Converse of 47

The first few and selected larger members of the sequence of factorials . The values specified in scientific notation are rounded to the displayed precision.

The first few amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368) .