GEM1518K - Mathematics in Art & Architecture - Project Submission. “For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art.” - M.C. Escher Contents Page 1 Introduction The Art of Alhambra Our Area of Focus Our Aim Page 2 The Principals behind Tessellations - Translation - Rotation - Reflection - Glide Reflection Page 3 Mathematics in Escher's Art - Translation - Rotation - Glide Reflection - Combination Page 4 Our Original Tessellations - The Catch (Translation) - Under the Sea (Rotation) - The Herd (Glide Reflection) - Dumbo & Butterfly (Combination) Page 5 Possible links with Architecture Conclusion References Mathematics in Escher's Art In this page we attempt to try to use simple mathematical terms to explain a few of Escher's pieces.
One of the basic principals behind his tessellations is the use of what we call an "addition and subtraction" method within the grid. Translation One of Escher's first explorations into the tessllations is that of the use of translation. Rotation Glide Reflection Combination. All time best Math video ever. Moebius transformation visualized in higher dimension | Talk Like A Physicist. Spiral of Theodorus. The spiral of Theodorus up to the triangle with a hypotenuse of √17 In geometry, the spiral of Theodorus (also called square root spiral, Einstein spiral or Pythagorean spiral)[1] is a spiral composed of contiguous right triangles. It was first constructed by Theodorus of Cyrene. History[edit] Plato does not attribute the irrationality of the square root of 2 to Theodorus, because it was well known before him.
Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories.[3] Hypotenuse[edit] Each of the triangles' hypotenuses hi gives the square root of the corresponding natural number, with h1 = √2. Plato, tutored by Theodorus, questioned why Theodorus stopped at √17. Overlapping[edit] Extension[edit] Colored extended spiral of Theodorus with 120 triangles Theodorus stopped his spiral at the triangle with a hypotenuse of √17.
Growth rate[edit] Angle[edit] If φn is the angle of the nth triangle (or spiral segment), then: with A triangle or section of spiral. Stereographic projection. 3D illustration of a stereographic projection from the north pole onto a plane below the sphere Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis, cartography, geology, and photography. In practice, the projection is carried out by computer or by hand using a special kind of graph paper called a stereographic net, shortened to stereonet or Wulff net. History[edit] Illustration by Rubens for "Opticorum libri sex philosophis juxta ac mathematicis utiles", by François d'Aiguillon. The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians.
Definition[edit] This section focuses on the projection of the unit sphere from the north pole onto the plane through the equator. Properties[edit] with E. Euler's formula. This article is about Euler's formula in complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic. Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.
Euler's formula states that, for any real number x, Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics. "[2] History[edit] It was Johann Bernoulli who noted that[3] And since the above equation tells us something about complex logarithms. Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand complex logarithms. Meanwhile, Roger Cotes, in 1714, discovered that ("ln" is the natural logarithm with base e).[4] where the real part the imaginary part. S&P :: Archive. 'Periodic table of shapes' to give a new dimension to math. Mathematicians are creating their own version of the periodic table that will provide a vast directory of all the possible shapes in the universe across three, four and five dimensions, linking shapes together in the same way as the periodic table links groups of chemical elements.
The three-year project should provide a resource that mathematicians, physicists and other scientists can use for calculations and research in a range of areas, including computer vision, number theory, and theoretical physics. The researchers, from Imperial College London and institutions in Australia, Japan and Russia, are aiming to identify all the shapes across three, four and five dimensions that cannot be divided into other shapes. As these building block shapes are revealed, the mathematicians will work out the equations that describe each shape and through this, they expect to develop a better understanding of the shapes' geometric properties and how different shapes are related to one another. Touch Trigonometry.
Trig-paperjs. Dave's short course in trigonometry. Table of Contents Who should take this course? Trigonometry for you Your background How to learn trigonometry Applications of trigonometry Astronomy and geography Engineering and physics Mathematics and its applications What is trigonometry? Trigonometry as computational geometry Angle measurement and tables Background on geometry The Pythagorean theorem An explanation of the Pythagorean theorem Similar triangles Angle measurement The concept of angle Radians and arc length Exercises, hints, and answers About digits of accuracy Chords What is a chord? Ptolemy’s sum and difference formulas Ptolemy’s theorem The sum formula for sines The other sum and difference formulas Summary of trigonometric formulas Formulas for arcs and sectors of circles Formulas for right triangles Formulas for oblique triangles Formulas for areas of triangles Summary of trigonometric identities More important identities Less important identities Truly obscure identities About the Java applet.
Warning! High Dimensions Ahead ← Inductio Ex Machina. Machine learning is often framed as finding surfaces (usually planes) that separate or fit points in some high dimensional space. Two and three dimensional diagrams are often used as an aid to intuition when thinking about these higher dimensional spaces. The following result (related to me by Marcus Hutter) shows that our low dimensional intuition can easily lead us astray.
Consider the diagram in Figure 1. The larger grey spheres have radius 1 and are each touching exactly two other grey spheres. The blue circle is then placed so that it touches all four grey circles. Pythagoras tells us that and so . Figure 1: The blue circle in the middle touches each of the four identical grey circles. We can move to three dimensions and set up an analogue of the two dimensional situation by arranging eight spheres of radius 1 so they are touching exactly three other grey spheres. As in the case with circles, we can place a blue sphere inside the eight grey spheres so that it touches all of them.