In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, an equilateral triangle is equiangular, it is a regular polygon, so it is referred to as a regular triangle. Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that: The area is A = 3 4 a 2 The perimeter is p = 3 a The radius of the circumscribed circle is R = a 3 The radius of the inscribed circle is r = 3 6 a or r = R 2 The geometric center of the triangle is the center of the circumscribed and inscribed circles The altitude from any side is h = 3 2 a Denoting the radius of the circumscribed circle as R, we can determine using trigonometry that: The area of the triangle is A = 3 3 4 R 2 Many of these quantities have simple relationships to the altitude of each vertex from the opposite side: The area is A = h 2 3 The height of the center from each side, or apothem, is h 3 The radius of the circle circumscribing the three vertices is R = 2 h 3 The radius of the inscribed circle is r = h 3 In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors, the medians to each side coincide.
A triangle ABC that has the sides a, b, c, semiperimeter s, area T, exradii ra, rb, rc, where R and r are the radii of the circumcircle and incircle is equilateral if and only if any one of the statements in the following nine categories is true. Thus these are properties that are unique to equilateral triangles, knowing that any one of them is true directly implies that we have an equilateral triangle. A = b = c 1 a + 1 b + 1 c = 25 R r − 2 r 2 4 R r s = 2 R + r s 2 = 3 r 2 + 12 R r s 2 = 3 3 T s = 3 3 r s = 3 3 2 R A = B = C = 60 ∘ cos A + cos B + cos C = 3 2 sin A 2 sin B 2 sin C 2 = 1 8 T = a 2 + b 2 + c 2 4 3 T = 3 4 2 3 R = 2 r 9 R 2 = a 2 + b 2 + c 2 r = r a +
The Koch snowflake is a mathematical curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch; as the fractal evolves, the area of the snowflake converges to 8/5 the area of the original triangle, while the perimeter of the snowflake diverges to infinity. The snowflake has a finite area bounded by an infinitely long line; the Koch snowflake can be constructed by starting with an equilateral triangle recursively altering each line segment as follows: divide the line segment into three segments of equal length. Draw an equilateral triangle that has the middle segment from step 1 as its base and points outward. Remove the line segment, the base of the triangle from step 2; the first iteration of this process produces the outline of a hexagram. The Koch snowflake is the limit approached; the Koch curve described by Helge von Koch is constructed using only one of the three sides of the original triangle.
In other words, three Koch curves make a Koch snowflake. A Koch curve–based representation of a nominally flat surface can be created by segmenting each line in a sawtooth pattern of segments with a given angle; each iteration multiplies the number of sides in the Koch snowflake by four, so the number of sides after n iterations is given by: N n = N n − 1 ⋅ 4 = 3 ⋅ 4 n. If the original equilateral triangle has sides of length s, the length of each side of the snowflake after n iterations is: S n = S n − 1 3 = s 3 n; the perimeter of the snowflake after n iterations is: P n = N n ⋅ S n = 3 ⋅ s ⋅ n. The Koch curve has an infinite length, because the total length of the curve increases by a factor of 4/3 with each iteration; each iteration creates four times as many line segments as in the previous iteration, with the length of each one being 1/3 the length of the segments in the previous stage. Hence, the length of the curve after n iterations will be n times the original triangle perimeter and is unbounded, as n tends to infinity.
As the number of iterations tends to infinity, the limit of the perimeter is: lim n → ∞ P n = lim n → ∞ 3 ⋅ s ⋅ n = ∞, since |4/3| > 1. An ln 4/ln 3-dimensional measure has not been calculated so far. Only upper and lower bounds have been invented. In each iteration a new triangle is added on each side of the previous iteration, so the number of new triangles added in iteration n is: T n = N n − 1 = 3 ⋅ 4 n − 1 = 3 4 ⋅ 4 n; the area of each new triangle added in an iteration is 1/9 of the area of each triangle added in the previous iteration, so the area of each triangle added in iteration n is: a n = a n − 1 9 = a 0 9 n. where a0 is the area of the original triangle. The total new area added in iteration n is therefore: b n = T n ⋅ a n = 3 4 ⋅ n ⋅ a 0 The total area of the snowflake after n iterations is: A n = a 0 + ∑ k = 1 n b k = a 0 = a 0 ( 1 + 1 3 ∑ k = 0 n − 1
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, C is denoted △ A B C. In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. In other words, there is only one plane that contains that triangle, every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; this article is about triangles in Euclidean geometry, in particular, the Euclidean plane, except where otherwise noted. Triangles can be classified according to the lengths of their sides: An equilateral triangle has all sides the same length. An equilateral triangle is a regular polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length. An isosceles triangle has two angles of the same measure, namely the angles opposite to the two sides of the same length; some mathematicians define an isosceles triangle to have two equal sides, whereas others define an isosceles triangle as one with at least two equal sides.
The latter definition would make all equilateral triangles isosceles triangles. The 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths. Equivalently, it has all angles of different measure. Hatch marks called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of short line segments in the form of tally marks. In a triangle, the pattern is no more than 3 ticks. An equilateral triangle has the same pattern on all 3 sides, an isosceles triangle has the same pattern on just 2 sides, a scalene triangle has different patterns on all sides since no sides are equal. Patterns of 1, 2, or 3 concentric arcs inside the angles are used to indicate equal angles. An equilateral triangle has the same pattern on all 3 angles, an isosceles triangle has the same pattern on just 2 angles, a scalene triangle has different patterns on all angles since no angles are equal.
Triangles can be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°; the side opposite to the right angle is the longest side of the triangle. The other two sides are called the catheti of the triangle. Right triangles obey the Pythagorean theorem: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 + 42 = 52. In this situation, 3, 4, 5 are a Pythagorean triple; the other one is an isosceles triangle. Triangles that do not have an angle measuring 90° are called oblique triangles. A triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle.
If c is the length of the longest side a2 + b2 > c2, where a and b are the lengths of the other sides. A triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side a2 + b2 < c2, where a and b are the lengths of the other sides. A triangle with an interior angle of 180° is degenerate. A right degenerate triangle has collinear vertices. A triangle that has two angles with the same measure has two sides with the same length, therefore it is an isosceles triangle, it follows that in a triangle where all angles have the same measure, all three sides have the same length, such a triangle is therefore equilateral. Triangles are assumed to be two-dimensional plane figures. In rigorous treatments, a triangle is therefore called a 2-simplex. Elementary facts about triangles were presented by Euclid in books 1–4 of his Elements, around 300 BC; the sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees.
This fact is equivalent to Euclid's parallel postulate. This allows determination of the measure of the third angle of any triangle given the measure of two angles. An exterior angle of a triangle is an angle, a linear pair to an interior angle; the measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it. The sum of the measures of the three exterior angles of any triangle is 360 degrees. Two triangles are said to be similar if every angle of one triangle has the same measure as the corresponding angle in the other triangle; the corresponding sides of similar triangles have lengths that are in the same proportion, this property is sufficient to establish similarity. Some basic theorems about similar triangles are: If and only if one pair of internal angles of two triangles have the sam
The pun called paronomasia, is a form of word play that exploits multiple meanings of a term, or of similar-sounding words, for an intended humorous or rhetorical effect. These ambiguities can arise from the intentional use of homophonic, metonymic, or figurative language. A pun differs from a malapropism in that a malapropism is an incorrect variation on a correct expression, while a pun involves expressions with multiple interpretations. Puns may be regarded as in-jokes or idiomatic constructions as their usage and meaning are specific to a particular language or its culture. Puns have a long history in human writing. For example, the Roman playwright Plautus was famous for word games. Puns can be classified in various ways; the homophonic pun, a common type, are not synonymous. Walter Redfern summarized this type with his statement, "To pun is to treat homonyms as synonyms." For example, in George Carlin's phrase "atheism is a non-prophet institution", the word prophet is put in place of its homophone profit, altering the common phrase "non-profit institution".
The joke "Question: Why do we still have troops in Germany? Answer: To keep the Russians in Czech" relies on the aural ambiguity of the homophones check and Czech. Puns are not homophonic, but play on words of similar, not identical, sound as in the example from the Pinky and the Brain cartoon film series: "I think so, but if we give peas a chance, won't the lima beans feel left out?" which plays with the similar—but not identical—sound of peas and peace in the anti-war slogan "Give Peace a Chance". A homographic pun exploits words which are spelled the same but possess different meanings and sounds; because of their nature, they rely on sight more than hearing, contrary to homophonic puns. They are known as heteronymic puns. Examples in which the punned words exist in two different parts of speech rely on unusual sentence construction, as in the anecdote: "When asked to explain his large number of children, the pig answered simply:'The wild oats of my sow gave us many piglets.'" An example that combines homophonic and homographic punning is Douglas Adams's line "You can tune a guitar, but you can't tuna fish.
Unless of course, you play bass." The phrase uses the homophonic qualities of tune a and tuna, as well as the homographic pun on bass, in which ambiguity is reached through the identical spellings of, and. Homographic puns do not need to follow grammatical rules and do not make sense when interpreted outside the context of the pun. Homonymic puns, another common type, arise from the exploitation of words which are both homographs and homophones; the statement "Being in politics is just like playing golf: you are trapped in one bad lie after another" puns on the two meanings of the word lie as "a deliberate untruth" and as "the position in which something rests". An adaptation of a joke repeated by Isaac Asimov gives us "Did you hear about the little moron who strained himself while running into the screen door?" Playing on strained as "to give much effort" and "to filter". A homonymic pun may be polysemic, in which the words must be homonymic and possess related meanings, a condition, subjective.
However, lexicographers define polysemes as listed under a single dictionary lemma while homonyms are treated in separate lemmata. A compound pun is a statement. In this case, the wordplay cannot go into effect by utilizing the separate words or phrases of the puns that make up the entire statement. For example, a complex statement by Richard Whately includes four puns: "Why can a man never starve in the Great Desert? Because he can eat the sand, there, but what brought the sandwiches there? Why, Noah sent Ham, his descendants mustered and bred." This pun uses sand, there/sandwiches there, Ham/ham, mustered/mustard, bred/bread. The phrase "piano is not my forte" links two meanings of the words forte and piano, one for the dynamic markings in music and the second for the literal meaning of the sentence, as well as alluding to "pianoforte", the older name of the instrument. Compound puns may combine two phrases that share a word. For example, "Where do mathematicians go on weekends? To a Möbius strip club!"
Puns on the terms Möbius strip club. A recursive pun is one in which the second aspect of a pun relies on the understanding of an element in the first. For example, the statement "π is only half a pie.". Another example is. Another example is "a Freudian slip is when you say one thing but mean your mother." The recursive pun "Immanuel doesn't pun, he Kant," is attributed to Oscar Wilde. Visual puns are sometimes used in logos, emblems and other graphic symbols, in which one or more of the pun aspects is replaced by a picture. In European heraldry, this technique is called canting arms. Visual and other puns and word games are common in Dutch gable stones as well as in some cartoons, such as Lost Consonants and The Far Side. Another type of visual pun exists in languages. For example, in Chinese, a pun may be based on a similarity in shape of the written character, despite a complete lack of phonetic similarity in the words punned upon. Mark Elvin describes how this "peculiarly Chinese form of visual punning involved comparing written characters to objects."
Richard J. Alexander notes two additional forms which puns may take: graphological (sometimes
A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to a variety of geometries. A periodic tiling has a repeating pattern; some special kinds include regular tilings with regular polygonal tiles all of the same shape, semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons; such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor or wall coverings.
Tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles. Decorative mosaic tilings made of small squared blocks called tesserae were employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made an early documented study of tessellations, he wrote about semiregular tessellations in his Harmonices Mundi. Some two hundred years in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries.
Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Aleksei Shubnikov and Nikolai Belov, Heinrich Heesch and Otto Kienzle. In Latin, tessella is a small cubical piece of stone or glass used to make mosaics; the word "tessella" means "small square". It corresponds to the everyday term tiling, which refers to applications of tessellations made of glazed clay. Tessellation in two dimensions called planar tiling, is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules; these rules can be varied. Common ones are that there must be no gaps between tiles, that no corner of one tile can lie along the edge of another; the tessellations created by bonded brickwork do not obey this rule. Among those that do, a regular tessellation has both identical regular tiles and identical regular corners or vertices, having the same angle between adjacent edges for every tile.
There are only three shapes that can form such regular tessellations: the equilateral triangle and regular hexagon. Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps. Many other types of tessellation are possible under different constraints. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner. Irregular tessellations can be made from other shapes such as pentagons, polyominoes and in fact any kind of geometric shape; the artist M. C. Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects. If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, these can be used to decorate physical surfaces such as church floors. More formally, a tessellation or tiling is a cover of the Euclidean plane by a countable number of closed sets, called tiles, such that the tiles intersect only on their boundaries.
These tiles may be any other shapes. Many tessellations are formed from a finite number of prototiles in which all tiles in the tessellation are congruent to the given prototiles. If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane; the Conway criterion is a sufficient but not necessary set of rules for deciding if a given shape tiles the plane periodically without reflections: some tiles fail the criterion but still tile the plane. No general rule has been found for determining if a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations. Mathematically, tessellations can be extended to spaces other than the Euclidean plane; the Swiss geometer Ludwig Schläfli pioneered this by defining polyschemes, which mathematicians nowadays call polytopes. These are the analogues to polygons and polyhedra in spaces with more dimensions, he further defined the Schläfli symbol notation to make it easy to describe polytopes.
For example, the Schläfli symbol for an equilateral triangle is. The Schläfli notation makes it possible to describe tilings compactly. For example, a tiling of regular hexagons has three six-sided polygons at each vertex, so its Schläfli symbol is. Other methods exist for describing polygonal tilings; when the tessellation
Asymmetry is the absence of, or a violation of, symmetry. Symmetry is an important property of both physical and abstract systems and it may be displayed in precise terms or in more aesthetic terms; the absence of or violation of symmetry that are either expected or desired can have important consequences for a system. Due to how cells divide in organisms, asymmetry in organisms is usual in at least one dimension, with biological symmetry being common in at least one dimension. Louis Pasteur proposed that biological molecules are asymmetric because the cosmic forces that preside over their formation are themselves asymmetric. While at his time, now, the symmetry of physical processes are highlighted, it is known that there are fundamental physical asymmetries, starting with time. Asymmetry is an important and widespread trait, having evolved numerous times in many organisms and at many levels of organisation. Benefits of asymmetry sometimes have to do with improved spatial arrangements, such as the left human lung being smaller, having one fewer lobes than the right lung to make room for the asymmetrical heart.
In other examples, division of function between the right and left half may have been beneficial and has driven the asymmetry to become stronger. Such an explanation is given for mammal hand or paw preference, an asymmetry in skill development in mammals. Training the neural pathways in a skill with one hand may take less effort than doing the same with both hands. Nature provides several examples of handedness in traits that are symmetric; the following are examples of animals with obvious left-right asymmetries: Most snails, because of torsion during development, show remarkable asymmetry in the shell and in the internal organs. Fiddler crabs have one small claw; the narwhal's tusk is a left incisor which can grow up to 10 feet in length and forms a left-handed helix. Flatfish have evolved to swim with one side upward, as a result have both eyes on one side of their heads. Several species of owls exhibit asymmetries in the size and positioning of their ears, thought to help locate prey. Many animals have asymmetric male genitalia.
The evolutionary cause behind this is, in most cases, still a mystery. Certain disturbances during the development of the organism, resulting in birth defects. Injuries after cell division that cannot be biologically repaired, such as a lost limb from an accident. Since birth defects and injuries are to indicate poor health of the organism, defects resulting in asymmetry put an animal at a disadvantage when it comes to finding a mate. In particular, a degree of facial symmetry is associated with physical attractiveness, but complete symmetry is both impossible and unattractive. Pre-modern architectural styles tended to place an emphasis on symmetry, except where extreme site conditions or historical developments lead away from this classical ideal. To the contrary and postmodern architects became much more free to use asymmetry as a design element. While most bridges employ a symmetrical form due to intrinsic simplicities of design and fabrication and economical use of materials, a number of modern bridges have deliberately departed from this, either in response to site-specific considerations or to create a dramatic design statement.
Some asymmetrical structures In fire-resistance rated wall assemblies, used in passive fire protection, but not limited to, high-voltage transformer fire barriers, asymmetry is a crucial aspect of design. When designing a facility, it is not always certain, that in the event of fire, which side a fire may come from. Therefore, many building codes and fire test standards outline, that a symmetrical assembly, need only be tested from one side, because both sides are the same. However, as soon as an assembly is asymmetrical, both sides must be tested and the test report is required to state the results for each side. In practical use, the lowest result achieved is the one. Neither the test sponsor, nor the laboratory can go by an opinion or deduction as to which side was in more peril as a result of contemplated testing and test only one side. Both must be tested in order to be compliant with test standards and building codes There are no a and b such that a < b and b < a. This form of asymmetry is an asymmetrical relation.
Certain molecules are chiral. Chemically identical molecules with different chirality are called enantiomers. Asymmetry arises in physics in a number of different realms; the original non-statistical formulation of thermodynamics was asymmetrical in time: it claimed that the entropy in a closed system can only increase with time. This was using the Clausius' Theorem; the theory of statistical mechanics, however, is symmetric in time. Although it states that a system below maximum entropy is likely to evolve towards higher entropy, it states that such a system is likely to have evolved from higher entropy. Symmetry is one of the most powerful tools in particle physics, because it has become evident that all laws of nature originate in symmetries. Violations of symmetry therefore present theoretical and experimental puzzles that lead to
Solomon W. Golomb
Solomon Wolf Golomb was an American mathematician and professor of electrical engineering at the University of Southern California, best known for his works on mathematical games. Most notably, he coined the name, he fully described polyominoes and pentominoes in 1953. He specialized in problems of combinatorial analysis, number theory, coding theory, communications, his game of pentomino inspired Tetris. Golomb, a graduate of the Baltimore City College high school, received his bachelor's degree from Johns Hopkins University and master's and doctorate degree in mathematics from Harvard University in 1957 with a dissertation on "Problems in the Distribution of the Prime Numbers". While working at the Glenn L. Martin Company he became interested in communications theory and began his work on shift register sequences, he spent his Fulbright year at the University of Oslo and joined the Jet Propulsion Laboratory at Caltech, where he researched military and space communications. He was awarded full tenure two years later.
Golomb pioneered the identification of the characteristics and merits of maximum length shift register sequences known as pseudorandom or pseudonoise sequences, which have extensive military and consumer applications. Today, millions of cordless and cellular phones employ pseudorandom direct-sequence spread spectrum implemented with shift register sequences, his efforts made USC a center for communications research. Golomb was the inventor of a form of entropy encoding. Golomb rulers, used in astronomy and in data encryption, are named for him, as is one of the main generation techniques of Costas arrays, the Lempel-Golomb generation method, he was a regular columnist, writing Golomb's Puzzle Column in the IEEE Information Society Newsletter. He was a frequent contributor to Scientific American's Mathematical Games column and a frequent participant in Gathering 4 Gardner conferences. Among his contributions to recreational mathematics are Rep-tiles, he contributed a puzzle to each issue of the Johns Hopkins Magazine, a monthly publication of his undergraduate alma mater, for a column called "Golomb's Gambits", was a frequent contributor to Word Ways: The Journal of Recreational Linguistics.
Golomb was the National Academy of Science. In 1985, he received the Shannon Award of the Information Theory Society of the IEEE. In 1992, he received the medal of the U. S. National Security Agency for his research, has been the recipient of the Lomonosov Medal of the Russian Academy of Science and the Kapitsa Medal of the Russian Academy of Natural Sciences. In 2000, he was awarded the IEEE Richard W. Hamming Medal for his exceptional contributions to information sciences and systems, he was singled out as a major figure of coding and information theory for over four decades for his ability to apply advanced mathematics to problems in digital communications. Golomb was one of the first high profile professors to attempt the Ronald K. Hoeflin Mega IQ power test, which appeared in Omni Magazine, he scored at least IQ 176. In 2012, he became a fellow of the American Mathematical Society; that same year, it was announced. In 2014, he was elected as a fellow of the Society for Industrial and Applied Mathematics "for contributions to coding theory, data encryption and mathematical games."In 2013, he was awarded the National Medal of Science 2011.
In 2016, he was awarded the Benjamin Franklin Medal in Electrical Engineering "for pioneering work in space communications and the design of digital spread spectrum signals, transmissions that provide security, interference suppression, precise location for cryptography. Signal Design for Good Correlation Polyominoes, Princeton University Press. ISBN 0-89412-048-4 Golomb graph Golomb sequence Polyomino Solomon Golomb Biography of Dr. Golomb at the USC Electrical Engineering Department's website Solomon W. Golomb at the Mathematics Genealogy Project