Erica Watson MAT 410 - Geometries Dr. Powers College of Saint Rose 04-24-2014

Country: Greece

Contribution:

Promoted the study of mathematics as an abstract process, rather than a concrete one.

Probably the first to prove the Pythagorean Theorem, even though the relationship had likely been known for centuries prior.

Country: China

Contribution:

+Wrote the "Mo Jing," the oldest surviving book of Chinese geometry. It might be thought of as the Chinese analog to Euclid's "Elements" in certain respects.

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Country: Greece

Contribution:

+Developed a new theory of proportion that allowed comparison of irrational numbers, which had previously been impossible.

+Used an early form of integration to prove theorems involving the volumes of cones and pyramids.

Country: Greece/Egypt

Contribution:

+Wrote the book series "Elements" which became the most widely copied and translated geometry text for nearly 2000 years.

+Formalized the idea and method of proof in geometry.

Country: Greece/Sicily

Contribution:

+Discovered theorems about the center of gravity for plane figures and solids.

+Wrote many books on geometry, including whole volumes on spirals, spheres/cylinders, conoids/spheroids, and measuring circles.

+Wrote a book describing how he achieved his discoveries. Many were apparently the result of mechanical experiment and observation, which he then backed up by geometric proof. (Sound familiar? See-saw? Thought so!)

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Country: Greece

Contribution:

+Discovered the Conchoid Curve, which can be used to trisect angles

Country: Greece

Contribution:

+Measured the Earth's circumference, and tilt of its axis, to a remarkable degree of accuracy for the time.

Country of Origin: Greece

Contribution:

+Wrote the book "Conics"

+Coined terms "parabola", "ellipse", and "hyperbola"

Country: Greece/Egypt

Contribution:

+Discovered a formula for calculating the area of a triangle given only the lengths of it's sides.

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Country: Greece/Egypt

Contribution:

+Studied spherical geometry extensively. In fact he was the first to put the definition of a spherical triangle in writing. He also made improvements in proofs in this field.

+Used spherical geometry in his study of astronomy.

+Discovered the relationship of collinear points on a triangles sides, described in "Menelaus' Theorem."

Country: Greece/Egypt

Contribution:

+Discovered his own Finite geometry, which has 9 points, 9 lines, and which allows for parallelism.

+Discovered a theorem which helped form the field of projective geometry, which did not become a serious field of study until more than 1000 years after his death.

Country: Greece/Egypt

Contribution:

+Among the first "mothers of mathematics."

+Wrote a commentary on Apollonius' "Conics."

+Edited a commentary, written by her father, on Euclid's "Elements."

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Country: China

Contribution:

+Devised a method of computing the diameter of a sphere given its volume.

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Country: China

Contributions:

+Wrote the book "Jigu Suanjing" ("Continuation of Ancient Mathematics") which contains 20 problems involving astronomical motion, engineering, volume, and right triangles. He solves them numerically using cubic equations, and describes this solution not using exponents, but in a distinctly geometric way, referencing the sides of a cube.

Country: India

Contribution:

+Discovered a formula for the area of any cyclic quadrilateral.

Country: Turkey/Iraq

Contribution:

+Was the first to introduce and use arithmetic operations with geometric quantities, which had previously been viewed by the Greeks as strictly non-numerical.

+Generalized the Pythagorean Theorem to arbitrary triangles (although Pappus had done this before him.)

Country: Iran/Iraq

Contribution:

+Wrote a book for craftsmen on geometric constructions, many of which he created using a ruler and fixed compass, probably because of its greater accuracy.

+During a long study of the Moon's orbit, he has the first recorded use of the Tangent function, made a table of the values of the Sine and Tangent functions at 15 degree intervals (which are accurate to 8 decimal places), and introduced the functions Secant and Cosecant.

Country: Iraq/Egypt

Contribution:

+One of the developers of analytic geometry, and the link between algebra and geometry.

+Discovered what would later be the "Lambert quadrilateral" and his theorems about it became the first theorems in elliptical and hyperbolic geometry.

Country: Iran

Contribution:

+While seeking to prove Euclid's 5th postulate, he inadvertently proved aspects of non-euclidean geometric figures.

Country: Italy

Contribution:

+Created the Fibonacci sequence, which has significant applications in geometry.

+Wrote "Practica Geometriae" largely based on the work of Euclid. It includes theorems and proofs, a method involving similar triangles for calculating the height of objects,as well as work on circumscribed polygons and polygons with circles inscribed in them.

Country: Italy

Contribution:

+Wrote "Geometry" which was a Latin version of Euclid's "Elements"

+Wrote "De Divina Proportione" which is a book on proportion and perspective in art, architecture, and mathematics, particularly focused on the Golden Ratio.

Country: Italy

Contributions:

+Used his understanding of mechanics to solve geometric problems, including mechanical solutions to "squaring the circle"

+Drew a series of figures for Luca Pacioli's book "De Divina Propotione." The images were of regular solids in "skeletal" or "wire-frame" style.

Country: Italy

Contribution:

+Used geometry to verify the Copernican theory that the Earth is not stationary, but revolves around the sun. In his book "Il Saggiatore", he makes it clear that geometry is at the heart of his work saying, "It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word..."

Country: Germany

Contribution:

+Wrote "Mysterium Cosmographicum" in which he discovered a relationship between the orbits of known planets and the Plantonic solids. It was a very accurate model for its time, closely corresponding to available astronomical data.

+Worked with the emerging discipline of projective geometry, explaining how conic sections could morph by moving their foci, and introduced an idea that would later be used by many other mathematician and come to be known as geometric continuity, aka the Law/Principle of Continuity.

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Country: France

Contributions:

+Worked with Conic Sections, simplifying some of the proofs of Apollonius, and introducing the idea of deformation of figures, such as how to deform a circle into an ellipse.

+Found a way to transform a square into another polygon of equal area with an arbitrary number of sides.

Country: France

Contribution:

+Worked extensively with theories of perspective and conic sections.

+Considered one of the founders of projective Geometry.

+Created his own finite geometry, the most famous theorem of which says that if two triangles are in perspective, then the vertices of their corresponding sides are collinear.

Country: France

Contributions:

+Invented the Cartesian coordinate system.

+Invented the x, y, and z notation for unknowns, as well as the a, b, and c notation for constants.

+Invented the superscript notation for exponents.

+Developed the field of analytic geometry.

Country: France

Contributions:

+At age 16, after studying the work of Desargues, discovered several projective geometry theorems of his own.

+Wrote "Essay on Conic Section" at age 17.

+Wrote an essay "De l'Esprit geometrique," which argued for the geometric means of finding truth (beginning from assumed principles and working forward logically) as a model for all philosophical thinking, so long as it was recognized that there was no way of knowing if the assumed principles were true.

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Country: Italy

Contribution:

+Rediscovered Menelaus' theorem regarding 3 collinear points

+Proved his own theorem regarding 3 concurrent lines

Country: Italy

Contribution:

+Wrote the book "Euclid Cleared of Every Defect", which is major work in non-euclidean geometry, which, oddly enough, came out of his attempt to prove Euclid's 5th postulate by contradiction.

Country: Germany

Contribution:

+Talked about "analysis situs" which was the forerunner of topology, a field that would not come into full study until more than 100 years later.

+Described any straight lines as a "curve, any part of which is similar to the whole" which is essentially a description of self-similarity, and is also an improvement over Euclid's idea of a straight line.

Country: Switzerland

Contribution: ALL THE THINGS!!!

+Introduced a lot of new notation, including the lowercase Greek letter for the number Pi, and Delta for change.

+Investigated surfaces and curvature, much of which was unpublished and found later by Gauss.

+Essentially invented the field of Graph Theory with his paper on the Konigsberg Bridges.

+Worked with polyhedra and found results that would later be important to topology.

Country: Switzerland

Contribution:

+Wrote the first proof that Pi is irrational.

+Did a study on the parallel postulate, assuming it was false, and came up with a new, non-euclidean geometry as a result.

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Country: France

Contribution:

+Came up with the first version of knot theory!

Country: England

+Offered yet another new parallel postulate, "There exists exactly one line parallel to a given line through a point not on the line."

Country: Germany

Contribution:

+Discovered a method for drawing a regular 17-sided polygon with a ruler and compass.

+Did major work in the field of Differential Geometry, discovering Gaussian curvature, which is extremely important in physics.

Country: Germany

Contribution:

+Wrote a book "Der barycentrische Calcul", which is primarily about analytic geometry, but also contains some of his work in projective and affine geometry, including his Mobius net, which became important in the further development of projective geometry.

+Discovered the Mobius strip, which he found while exploring a geometric theory about polyhedrons.

Country: Russia

Contribution:

+Offered yet another new parallel postulate, "There exist 2 lines parallel to a given line through a point not on the line."

Country: Germany

Contribution:

+Wrote one of the first textbooks on Topology.

+Discovered the Mobius strip about the same time as Mobius did, but the two were working independently.

Country: England

Contribution:

+Produced some 250 papers on math while working as a lawyer. He happily took a serious pay cut by giving up law to accept a professorship at Cambridge in their math department. (One does not go into math for money, but love.)

+Combined projective and metric geometry.

+Worked on non-euclidean and n-dimensional geometry.

+Stated that Euclidean Geometry was "absolutely true" in Euclidean space, rather than only "approximately true" because it was becoming apparent that the physical space of our experience was only "approximately Euclidean."

Country: Germany

Contribution:

+His thesis looked at geometry of analytic functions as well as the connectivity of surfaces. In it he talks about surfaces in the complex plane, now called Riemann Surfaces, which are important to differential geometry.

+His definition of n-dimensional space became the modern Riemannian space. Both this and his definition of the curvature tensor are key elements in differential geometry.

Country: England

Contribution:

+Author of the novella "Flatland," which was used to help explain the concept of 4D space.

Country: Germany

Contribution:

+One of the Founders of Algebraic Geometry

+Father of Emmy Noether (She is my personal hero! Thank you good sir!)

Country: Germany

Contribution:

+Showed that non-euclidean geometry was consistent if and only if euclidean geometry was consistent. This removed any remaining controversy about the validity of the field of non-euclidean geometry. It is the reason why we now look at geometry as consisting of both subtypes.

Country: France

Contributions:

Original discoverer of chaos theory while working on the "three-body problem."

Country: Ireland

Contribution:

+Extensive work in 4D geometry, coined the term "polytope" to refer to a convex solid in 4 dimensions.

+Used Euclidean constructions to create 3D cross sections of the 6 regular 4D polytopes she discovered.

Country: Germany

Contribution:

+Wrote the textbook "Grundlagen der Geometrie"("Foundations of Geometry"), in which he reworked many of Euclid's geometric axioms.

Country: Germany

Contribution:

+One of the founders of modern topology

+Came up with the concept of the "Hausdorff dimension," which is important for understanding certain aspects of fractals.

Country: Germany

Contribution:

+Discovered the "Koch curve" which is a fractal. It is continuous but completely non-differentiable.

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Country: USA

Contribution:

+Created Young's Geometry, a finite geometry similar to Fano's, except that Young changed his last axiom so that the system allows for parallelism.

Country: USA

Contribution:

+Worked with J.W. Young to prove an important theorem for projective geometry.

+Proved the Jordan Curve Theorem

Country: Poland

Contribution:

+Created a number of fractals, including the Sierpinski Curve, the Sierpinski Carpet, and the Sierpinski Triangle (Gasket).

Country: Netherlands

Contributions:

+Extensive work with tessellations and "impossible" constructions, symmetry, and other mathematical concepts in his art work.

+His work shows the profound beauty to be found in mathematics as an art like few before or since.

Country: England/Canada

Contribution:

+Worked on polytopes and group theory, and discovered something called Coxeter groups, which produce tessellations.

Country: Japan

Contribution:

+Extensive work in the field of Differential Geometry, but also worked on a huge variety of other geometries and related subjects.

Country: Hungary

Contribution:

+Wrote several books, including one called "Regular Figures" which covers various Polyhedra, hyperbolic tessellations, spherical arrangements, etc.

+Coined the term "Intuitive Geometry" which he meant to refer to geometry which was accessible to everyone. It encompasses packing theory, tiling, convexity, crystallography, and the geometry of numbers, among other things

Country: USA

Contribution:

+(Re)Discovered the chaos theory of Poincare. Also found "strange attractors," which are critical to the study of fractals.

Country: France/USA

Contribution:

+Discovery of the Mandelbrot Set

+Advancement of Chaos Theory and fractals

Country: France

Contribution:

+Works extensively with algebraic geometry, and has written a number of books on it.

+Was one of the later members of the "Nicolas Bourbaki" group.

Country: USA

Contribution: Helped prove the 4-Color Theorem

Country: Japan

Contributions:

+Created a geometric model of number theory

+Extensive work with hypothetical crystals using discrete geometry led him to find the "K_4 crystal," which he called the "mathematical twin" of the diamond crystal.

Country: China/USA

Contribution:

+Extensive work on chaos theory, particularly with strange attractors, and she continues to study the fractal dimension.