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The Egyptian pyramids were built using geometry, with careful measurements to make sure they were perfectly shaped and aligned. They used math to create the pyramid's straight sides and precise angles, making them stable and impressive even today.

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Heron's Formula, developed around 2,000 BC, is a way to calculate the area of a triangle when you know the lengths of all three sides. It uses geometry by applying the semi-perimeter (half the perimeter) and the side lengths to find the area without needing to know the height.

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The Moscow Papyrus, dating back to around 1800 BC, is an ancient Egyptian document that contains mathematical problems, including methods for calculating areas of various shapes. It uses geometry to solve practical problems, such as finding the area of a circle, by approximating the value of pi and applying formulas to measure land and buildings.

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The Rhind Papyrus, created around 1650 BC, is an ancient Egyptian mathematical text that includes a variety of arithmetic and geometric problems. It uses geometry to calculate areas of shapes like triangles, rectangles, and circles, and demonstrates early methods for solving practical problems in surveying and construction.

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The square root of 2, known since around 800 BC, was important in ancient geometry, particularly in finding the diagonal of a square. Ancient mathematicians used geometry to approximate the square root of 2 by constructing right triangles, applying the Pythagorean theorem to find the length of the diagonal when the sides of the square are of unit length.

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The Shatapatha Brahmana, composed around 700 BC, is an ancient Indian text that contains religious hymns, rituals, and also mathematical concepts. It uses geometry in the form of precise measurements and instructions for constructing altars, where geometric principles such as the use of squares, rectangles, and circular patterns were applied to align and design sacred spaces for rituals.

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Brahmagupta's Formula, written in 628 AD, provides a way to calculate the area of a cyclic quadrilateral (a four-sided figure with vertices on a circle) using the lengths of its sides. This formula applies geometry by linking the side lengths to the area, demonstrating an early understanding of geometric relationships and providing a practical tool for architects and engineers of the time.

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Pythagorean triples, known around 600 BC, are sets of three whole numbers that satisfy the Pythagorean theorem, where the sum of the squares of two sides of a right triangle equals the square of the hypotenuse. These triples are used in geometry to find whole-number solutions for the side lengths of right-angled triangles, helping ancient mathematicians and builders create precise measurements in their constructions.

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Grigori Perelman solves the Poincaré Conjecture, a landmark result in topology, which was also deeply connected to geometric ideas about the structure of three-dimensional spaces.

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Benoît B. Mandelbrot develops the theory of fractals, discovering geometric shapes that exhibit self-similarity at different scales and have important implications in nature, art, and mathematics.

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With the advent of computers, computational geometry begins to develop as a field, particularly in the application of geometry to computer graphics and algorithms.

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Projective geometry, a field that explores geometric properties invariant under projection, gains significant traction with mathematicians like Giuseppe Peano and others.

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The Development of Differential Geometry 

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The Development of Differential Geometry 

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The work of mathematicians like Élie Cartan and others solidifies the importance of differential geometry, influencing modern physics, particularly in general relativity.

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The First International Congress of Mathematicians

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The First International Congress of Mathematicians

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The Congress solidifies the growing interest in geometry, particularly in areas like non-Euclidean geometry and topology.

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The First International Congress of Mathematicians

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Minkowski Space and Geometry of Spacetime

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Minkowski Space and Geometry of Spacetime

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Hermann Minkowski develops a geometric interpretation of Einstein’s special theory of relativity by using a four-dimensional space, later known as Minkowski space.

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Minkowski Space and Geometry of Spacetime

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Poincaré develops topology as a branch of mathematics, beginning with the study of the properties of geometric objects that are preserved under continuous transformations.

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Felix Klein presents his Erlangen Program, showing that geometry can be understood as the study of groups of transformations, fundamentally linking algebra and geometry.

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William Rowan Hamilton Develops Quaternions

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William Rowan Hamilton Develops Quaternions

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Hamilton formulates quaternions, a system of numbers used to extend the concept of complex numbers to higher dimensions, providing a new way to approach geometric transformations in 3D space.

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William Rowan Hamilton Develops Quaternions

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Bernhard Riemann introduces Riemannian geometry, a form of geometry that generalizes Euclidean geometry and forms the basis for Einstein’s theory of general relativity.

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Lobachevsky and Bolyai Develop Hyperbolic Geometry

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Lobachevsky and Bolyai Develop Hyperbolic Geometry

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Both mathematicians independently develop hyperbolic geometry, showing that the parallel postulate does not hold in their new geometries.

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Lobachevsky and Bolyai Develop Hyperbolic Geometry

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Carl Friedrich Gauss' Work on Non-Euclidean Geometry

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Carl Friedrich Gauss' Work on Non-Euclidean Geometry

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Gauss investigates the possibility of non-Euclidean geometry, laying the groundwork for later developments by Lobachevsky and Bolyai.

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Lambert proves the existence of non-Euclidean geometries, showing that the parallel postulate does not necessarily hold in all geometrical frameworks.

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Leonhard Euler makes significant contributions to topology, graph theory, and geometry, including the Euler characteristic and Euler's polyhedron formula (V - E + F = 2).

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Bonaventura Cavalieri develops the method of indivisibles, an early precursor to integral calculus, allowing for the calculation of areas and volumes.

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Newton uses geometry extensively in formulating the laws of motion and gravitation, particularly in his geometric approach to the law of universal gravitation.

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Fermat and Descartes' Contributions to Geometry

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Fermat and Descartes' Contributions to Geometry

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Fermat introduces methods of finding tangents to curves and the idea of optimizing geometric shapes. Descartes' work on coordinates provides a bridge between algebra and geometry.

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René Descartes establishes the connection between algebra and geometry through the development of analytic geometry, introducing Cartesian coordinates and the concept of graphs.

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Artists like Brunelleschi and Alberti develop the concept of linear perspective, a geometrical method for representing three-dimensional objects on a two-dimensional surface.

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Fibonacci introduces the Hindu-Arabic numeral system to Europe, influencing mathematical and geometric calculations.

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Arabic mathematical works, including those of Al-Khwarizmi, al-Battani, and others, are translated into Latin, influencing the European Renaissance and the development of geometry in the West.

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Al-Khwarizmi's "Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala

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Al-Khwarizmi's "Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala

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The Persian mathematician Al-Khwarizmi contributes to the development of algebra, which would later influence geometric reasoning, especially through his work on solving quadratic equations.