A Brief Timeline of World Geometers

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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.)

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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.

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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.

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

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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.

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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.

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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.

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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..."

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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!

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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."

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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.

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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.

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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."

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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.

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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."

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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.

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

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Country: Germany
Contribution:
+One of the Founders of Algebraic Geometry
+Father of Emmy Noether (She is my personal hero! Thank you good sir!)

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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.

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Country: France
Contributions:
Original discoverer of chaos theory while working on the "three-body problem."

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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.

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

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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.

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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.

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Country: USA
Contribution:
+Worked with J.W. Young to prove an important theorem for projective geometry.
+Proved the Jordan Curve Theorem

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

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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.

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Country: England/Canada
Contribution:
+Worked on polytopes and group theory, and discovered something called Coxeter groups, which produce tessellations.

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

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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

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

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Country: France/USA
Contribution:
+Discovery of the Mandelbrot Set
+Advancement of Chaos Theory and fractals

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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.

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Country: USA
Contribution: Helped prove the 4-Color Theorem

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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.

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Country: China/USA
Contribution:
+Extensive work on chaos theory, particularly with strange attractors, and she continues to study the fractal dimension.