Physical World/Mathematics - Revision history

QRNeditor: Added diagram of Platonic Solids

2022-11-28T22:13:08Z

Added diagram of Platonic Solids

QRNeditor: Corrected some errors in Geometry section

2021-09-09T11:16:45Z

Corrected some errors in Geometry section

QRNeditor: Formatting edits

2021-05-06T17:43:44Z

Formatting edits

Admin: Created page with "Geometry Surface area of a '''sphere''' – 4 π r² Volume of a sphere – 4/3 π r³ Surface area of a '''cylinder''' (with top and bottom) – 2 π r (r + h) Volume of..."

2021-04-18T22:02:27Z

Created page with "Geometry Surface area of a '''sphere''' – 4 π r² Volume of a sphere – 4/3 π r³ Surface area of a '''cylinder''' (with top and bottom) – 2 π r (r + h) Volume of..."

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 In Euclidean geometry, a '''Platonic solid''' is a regular, convex polyhedron with congruent faces of regular polygons and the same number of faces meeting at each vertex. Five solids meet those criteria, and each is named after its number of faces
+[[File:Platonic Solids Composition 51.svg|center|thumb|600x600px]]
 '''Tetrahedron''' – any polyhedron having four plane faces

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 Frustrum – cone formed by the cutting of a plane parallel to the base
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 A polygon has a minimum of three sides
-Regular polygon – triangle, square, pentagon, hexagon, octagon etc.
+A '''regular polygon''' has all its sides the same length and all its interior/internal angles the same size and all its exterior/external angles the same size
 Sum of exterior angles of a polygon is 360<sup>o</sup>
-Area of a '''triangle''' = (base x height) / 2
+Internal angles of a '''pentagon''' add up to 540 degrees. Each interior angle of a regular pentagon is 108 degrees
-In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. Each median of a triangle passes through the triangle's centroid
+Internal angles of a '''hexagon''' add up to 720 degrees. Each interior angle of a regular hexagon is 120 degrees
 Heptagon – 7-sided polygon
-Internal angles of an '''octagon''' add up to 1080 degrees. Each interior angle is 135 degrees
+Internal angles of an '''octagon''' add up to 1080 degrees. Each interior angle of a regular octagon is 135 degrees
 Nonagon – 9-sided polygon
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 '''Kite''', or deltoid, is a quadrilateral with two pairs of congruent adjacent sides
 A '''polyhedron''' is a solid in three dimensions with flat faces, straight edges and sharp corners or vertices
-Surface area of a '''pyramid''' – a<sup>2</sup> + 2 a h (where ‘a’ is side along the base)
+Surface area of a square based '''pyramid''' – a<sup>2</sup> + 2 a h (where ‘a’ is side along the base and ‘h’ is the triangular height)
-Volume of a pyramid – 1/3 a h
+Volume of a pyramid – 1/3 a h (Note that this is only true if ‘a’ is defined as the area of the base of the pyramid. Here ‘h’ is the vertical height)
 An n-sided pyramid is a polyhedron formed by connecting an n-sided polygonal base and a point, called the apex by n triangular faces (n ≥ 3). In other words, it is a conic solid with polygonal base
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 Circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices
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 '''Icosahedron''' has 20 identical equilateral triangular faces, 12 vertices and 30 edges
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 Eccentricity – a ratio describing the shape of a conic section; the ratio of the distance between the foci to the length of the major axis
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 Bezier curve – a parametric curve important in computer graphics. Used to design automobile bodies

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-Geometry
+== Geometry ==
 Surface area of a '''sphere''' – 4 π r²
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 Archimedean spiral (also known as the arithmetic spiral) is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity r = Aθ
+== Trigonometry ==
 Trigonometry – from the Greek trigonon ‘three angles’ and metro ‘measure’
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 Conjugate or explementary angles – two angles whose sum is 360°
+== Calculus ==
 Differential calculus – concerned with the instantaneous rate of change of quantities with respect to other quantities, or more precisely, the local behaviour of functions. This can be illustrated by the slope of a function's graph
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 Trapezium rule – an approximate technique for calculating the definite integral
+== Algebra ==
 Andrew Wiles solved '''Fermat’s Last Theorem''' in 1993. It states that it is impossible to find an integer solution to the equation x<sup>n</sup> + y<sup>n</sup> = z<sup>n</sup> if n is greater than 2 and x, y and z are non-zero. The proof was revised by Wiles in 1994 with the help of Richard Taylor
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 Subtraction formula. minuend (c) − subtrahend (b) = difference (a)
+== Numbers ==
 e and π are '''transcendental''' numbers, i.e. non-algebraic, cannot be expressed as solutions to a polynomial equation made up of rational numbers
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 Vinculum – a horizontal line used in mathematical notation, e.g. 0.3333 can be represented as 0.3¯
+== Number systems ==
 Long scale is the English translation of the French term echelle longue. It refers to a system of names in which every new term greater than million is 1,000,000 times the previous term: billion means a million to the power of two or a million millions (10<sup>12</sup>), trillion means a million to the power of three or a million billions (10<sup>18</sup>), and so on. Used in UK
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 Vigesimal – base 20 numeral system
+== Statistics ==
 '''Normal distribution''', also called the Gaussian distribution, is an important family of continuous probability distributions. Also known as a bell curve
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 Extreme value theory or extreme value analysis deals with the extreme deviations from the median of probability distributions. It seeks to assess the probability of events that are more extreme than any previously observed, e.g. a 100-year flood
+== Set theory ==
 Set theory is the branch of mathematical logic that studies sets, which are collections of objects
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 Euler diagram is a way of representing how sets overlap – similar to a Venn diagram. It is named after its inventor Leonhard Euler. Unlike a Venn diagram, it does not have to contain all possible zones (where a zone is defined as the area of intersection of the sets it shows)
+== Chaos theory ==
 Chaos theory is a study of behaviour of dynamical systems that are highly sensitive to initial conditions. A small variation in these initial conditions will produce wildly differing results. Chaos is a form of dynamical instability. This sensitivity is popularly referred to as the butterfly effect
 Edward Lorenz wrote ''Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?'', which led to the term ‘'''butterfly effect'''’
+== Graph theory ==
 Graph theory – the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A graph in this context is a collection of vertices or nodes and a collection of edges that connect pairs of vertices. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another
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 Four colour conjecture (four colour theorem) was proved by Wolfgang Haken and Kenneth Appel in 1976. Given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colours are required to colour the regions of the map so that no two adjacent regions have the same colour. It was the first major theorem to be proved using a computer
+== Game theory ==
 Game theory is the study of strategic decision making. Specifically, it is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers”
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 Zero-sum games – one person's gains exactly equal net losses of the other participant or participants
+== Prizes ==
 '''Fields Medal''' is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. Founded at the behest of Canadian mathematician John Charles Fields, the medal was first awarded in 1936, to Finnish mathematician Lars Ahlfors and American mathematician Jesse Douglas, and has been awarded every four years since 1950. On the obverse is Archimedes and a quote attributed to him written in Latin which means ‘Rise above oneself and grasp the world’