Clark Kimberling (born November 7, 1942, in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer sequences, and hymnology.
Kimberling received his PhD in mathematics in 1970 from the Illinois Institute of Technology, under the supervision of Abe Sklar. [1] Since at least 1994, he has maintained a list of triangle centers and their properties. In its current on-line form, the Encyclopedia of Triangle Centers, this list comprises tens of thousands of entries.
He has contributed to The Hymn, the journal of the Hymn Society in the United States and Canada; and in the Canterbury Dictionary of Hymnology .
Robert C. Schoen has defined a "golden triangle" as a triangle with two of its sides in the golden ratio. Kimberling has proposed that Schoen's definition of golden triangle be extended to include triangles which have angles that are in the golden ratio. [2] Kimberling has described a "doubly golden triangle" which has two sides that are in golden ratio and which also has two angles that are in golden ratio. [3]
In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes from 1 and 2. Starting from 0 and 1, the first few values in the sequence are:
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities and with ,
In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle △ABC, let the lines AO, BO, CO be drawn from the vertices to a common point O, to meet opposite sides at D, E, F respectively. Then, using signed lengths of segments,
In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider by a factor of φ for every quarter turn it makes.
In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, , which is , where is approximately 1.618.
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.
In geometry, a Heronian triangle is a triangle whose side lengths a, b, and c and area A are all positive integers. Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula which Heron demonstrated with the example triangle of sides 13, 14, 15 and area 84.
A Fibonacci word is a specific sequence of binary digits. The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition.
A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.
A golden triangle, also called a sublime triangle, is an isosceles triangle in which the duplicated side is in the golden ratio to the base side:
In Euclidean geometry, the isodynamic points of a triangle are points associated with the triangle, with the properties that an inversion centered at one of these points transforms the given triangle into an equilateral triangle, and that the distances from the isodynamic point to the triangle vertices are inversely proportional to the opposite side lengths of the triangle. Triangles that are similar to each other have isodynamic points in corresponding locations in the plane, so the isodynamic points are triangle centers, and unlike other triangle centers the isodynamic points are also invariant under Möbius transformations. A triangle that is itself equilateral has a unique isodynamic point, at its centroid(as well as its orthocenter, its incenter, and its circumcenter, which are concurrent); every non-equilateral triangle has two isodynamic points. Isodynamic points were first studied and named by Joseph Neuberg (1885).
The Fibonacci Quarterly is a scientific journal on mathematical topics related to the Fibonacci numbers, published four times per year. It is the primary publication of The Fibonacci Association, which has published it since 1963. Its founding editors were Verner Emil Hoggatt Jr. and Alfred Brousseau; the present editor is Professor Curtis Cooper of the Mathematics Department of the University of Central Missouri.
In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions.
A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is where is the golden ratio, and the progression can be written: , or approximately . Squares on the edges of this triangle have areas in another geometric progression, . Alternative definitions of the same triangle characterize it in terms of the three Pythagorean means of two numbers, or via the inradius of isosceles triangles.
In geometry, Napoleon points are a pair of special points associated with a plane triangle. It is generally believed that the existence of these points was discovered by Napoleon Bonaparte, the Emperor of the French from 1804 to 1815, but many have questioned this belief. The Napoleon points are triangle centers and they are listed as the points X(17) and X(18) in Clark Kimberling's Encyclopedia of Triangle Centers.
In geometry, the Spieker center is a special point associated with a plane triangle. It is defined as the center of mass of the perimeter of the triangle. The Spieker center of a triangle △ABC is the center of gravity of a homogeneous wire frame in the shape of △ABC. The point is named in honor of the 19th-century German geometer Theodor Spieker. The Spieker center is a triangle center and it is listed as the point X(10) in Clark Kimberling's Encyclopedia of Triangle Centers.
In mathematics, the Wythoff array is an infinite matrix of integers derived from the Fibonacci sequence and named after Dutch mathematician Willem Abraham Wythoff. Every positive integer occurs exactly once in the array, and every integer sequence defined by the Fibonacci recurrence can be derived by shifting a row of the array.
In Euclidean geometry, the Clawson point is a special point in a triangle defined by the trilinear coordinates tan α : tan β : tan γ, where α, β, γ are the interior angles at the triangle vertices A, B, C. It is named after John Wentworth Clawson, who published it 1925 in the American Mathematical Monthly. It is denoted X(19) in Clark Kimberling's Encyclopedia of Triangle Centers.
In mathematics, the fibbinary numbers are the numbers whose binary representation does not contain two consecutive ones. That is, they are sums of distinct and non-consecutive powers of two.
In mathematics, modern triangle geometry, or new triangle geometry, is the body of knowledge relating to the properties of a triangle discovered and developed roughly since the beginning of the last quarter of the nineteenth century. Triangles and their properties were the subject of investigation since at least the time of Euclid. In fact, Euclid's Elements contains description of the four special points – centroid, incenter, circumcenter and orthocenter - associated with a triangle. Even though Pascal and Ceva in the seventeenth century, Euler in the eighteenth century and Feuerbach in the nineteenth century and many other mathematicians had made important discoveries regarding the properties of the triangle, it was the publication in 1873 of a paper by Emile Lemoine (1840–1912) with the title "On a remarkable point of the triangle" that was considered to have, according to Nathan Altschiller-Court, "laid the foundations...of the modern geometry of the triangle as a whole." The American Mathematical Monthly, in which much of Lemoine's work is published, declared that "To none of these [geometers] more than Émile-Michel-Hyacinthe Lemoine is due the honor of starting this movement of modern triangle geometry". The publication of this paper caused a remarkable upsurge of interest in investigating the properties of the triangle during the last quarter of the nineteenth century and the early years of the twentieth century. A hundred-page article on triangle geometry in Klein's Encyclopedia of Mathematical Sciences published in 1914 bears witness to this upsurge of interest in triangle geometry.
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