Pages Needing Advanced Explanations

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The following image pages are in need of image explanations at an advanced level of mathematics (graduate level and beyond).

Anne Burns' MathscapesFractalsAnne M. BurnsIn her Mathscape images, Anne M. Burns combines recursive algorithms for clouds, mountains, and various imaginary plant forms into one picture.
Apollonian SnowflakeFractalsMe (Victor)
Apothems and AreaGeometryazavez1The image to the right shows the shortest distance from the center to the midpoint of one side in various regular polygons.
Application of the Euclidean AlgorithmNumber TheoryWouter Hisschemöller
ArbelosThis modern knife in the shape of an arbelos is used to make shoes.
Barnsley FernAlgebraMichael BarnsleyThe Barnsley Fern was created by Michael Barnsley using an iterated function system.
Basis of Vector SpacesAlgebraMathematicaThe same object, here a circle, can be completely different when viewed in other vector spaces.
Bedsheet ProblemTake a piece of paper. Now try to fold it in half more than 7 times. Is it possible? What is the ultimate number of folds a flat piece of material can achieve? This image shows Britney Gallivan’s success at folding a sheet 12 times.
Bezier CurvesAlgebraA Bezier Curve involves the use of two anchor points and a number of control points to control the form of a curve.
Blue WashFractalsPaul CockshottThis image is a random fractal that is created by continually dividing a rectangle into two parts and adjusting the brightness of each resulting part.
Bounding VolumesAlgebrachanjA box bounding the Stanford Bunny mesh.
BouquetGeometryGeorge W. HartThis is a 9-inch diameter table-top sculpture made of acrylic plastic (plexiglas). Bouquet has a very light and open feeling and gives very different impressions when viewed from different angles.
Boy's SurfaceGeometryPaul NylanderBoy's Surface was discovered in 1901 by German mathematician Werner Boy when he was asked by his advisor, David Hilbert, to prove that an immersion of the projective plane in 3-space was impossible. Today, a large model of Boy's Surface is displayed outside of the Mathematical Research Institute of Oberwolfach in Oberwolfach, Germany. The model was constructed as well as donated by Mercedes-Benz.
Boy's Surface VocabularyGeometryPaul NylanderWhile trying to prove that an immersion (a special representation) of the projective plane did not exist, German mathematician Werner Boy discovered Boy’s Surface in 1901. Boy’s Surface is an immersion of the projective plane in three-dimensional space. This object is a single-sided surface with no edges.
Bridge of PeaceAlgebraThe bridge of peace in Tbilisi ,Georgia, possesses a glass and steel covering frame which possesses a unique tiling structure, conic sections in its roof. Mapping a complicated pattern onto an uneven surface.
Broken HeartFractalsJos LeysA broken heart created by a variation on a fractal.
Brouwer Fixed Point TheoremTopologyRebecca
Brunnian LinksAlgebraRob SchareinThese are Borromean Rings...
Buffon's NeedleThe Buffon's Needle problem is a mathematical method of approximating the value of pi <math>(\pi = 3.1415...) </math>involving repeatedly dropping needles on a sheet of lined paper and observing how often the needle intersects a line.
Bump MappingAlgebraBump mapping is the process of applying a height map to a lit polygon to give a polygon the perception of depth.
Cantor SetTopologyKeith Peters
CardioidGeometryHenrik Wann JensenA Cardioid is a pattern defined by the path of a point of the circumference of a circle that rotates around another circle.
Catalan NumbersThis greedy little worm wants to eat the poor apple. He can only go to the east and to the north in this 8 by 8 grid. Since there is stain on the grid, he cannot pass above the diagonal connecting the worm and the apple. How many ways could he get there? The main image shows only one way of reaching the apple.
This is a very famous grid problem in combinatorics, which could be solved by Catalan numbers.
CatenaryA catenary is the curve created by a theoretical representation of a hanging chain or cable held at both ends.
Change Of Coordinate TransformationsAn example of various coordinate transformations applied to simple geometry.
Change of Coordinate SystemsThe same object, here a disk, can look completely different depending on which coordinate system is used.
ChryzodesNumber TheoryJ-F. Collonna &. J-P BourguignoChryzodes are visualizations of arithmetic using chords in a circle.
Circular Rotative Envelope IntersectionAlgebrak
CoefficientsAlgebraJust a quadratic function.
Compass & Straightedge Construction and the Impossible ConstructionsGeometryWikipediaThis image shows the step by step construction of a hexagon inscribed in the circle using a compass and a unmarked straightedge.
Conic SectionA conic section is a curve created from the intersection of a plane with a cone.
Controlling & Comparing The Blue Wash FractalAlgebraDifferent steps taken to control the Blue Wash Fractal on GSP. My goal was to iterate the rectangle so that it divides in half horizontally the first time and in half vertically the second time and so on. GSP was used to rotate the direction in which the rectangle is cut vertically and horizontally.
Cornu SpiralAlgebraThe Ponce de Leon Inlet Lighthouse is the tallest lighthouse in Florida. Its grand spiral staircase depicts the Cornu Spiral which is also commonly referred to the Euler Spiral.
Crop CirclesGeometryEiman EltiganiCrop circles, formed by crushed crops, are a pattern of geometric shapes, such as triangles, circles, etc. They illustrate many geometric theorems and relationships between the shapes of the pattern.
Dandelin Spheres TheoryGeometryHollister (Hop) David
Different StrokesFractalsLinda AllisonDifferent Strokes is generated with Ultra Fractal, a program designed by Frederik Slijkerman. It consists of 10 layers and uses both Julia and Mandelbrot fractal formulas and other formulas for coloring.
Dihedral GroupsEach snowflake in the main image has the dihedral symmetry of a natual regular hexagon. The group formed by these symmetries is also called the dihedral group of degree 6. Order refers to the number of elements in the group, and degree refers to the number of the sides or the number of rotations. The order is twice the degree.
Divergence TheoremThe water flowing out of a fountain demonstrates an important theorem for vector fields, the Divergence Theorem.
Dragons 1GeometryJos LeysA tessellation created in the style of M.C. Escher.
Dual PolyhedronGeometryMathWorld

This is a beautiful blue-aerial-shell firework filling the sky. Each particle of the firework follows a parabolic trajectory, and together they sweep an area with the red curve as its boundary. This red boundary is then called the envelope of those parabolas. What's more, as we are going to see in the following sections, this envelope also turns out to be a parabola.
EpitrochoidsGeometryAlbrecht DuererAn epitrochoid is a roulette made from a circle going around another circle. A roulette is a curve that is created by tracing a point attached to a rolling figure.
Euclidean AlgorithmAbout 2000 years ago, Euclid, one of the greatest mathematician of Greece, devised a fairly simple and efficient algorithm to determine the greatest common divisor of two integers, which is now considered as one of the most efficient and well-known early algorithms in the world. The Euclidean algorithm hasn't changed in 2000 years and has always been the the basis of Euclid's number theory. This image shows Euclid's method to find the greatest common divisor of two integers. The greatest common divisor of two numbers a and b is the largest integer that divides the numbers without a remainder.
Euler's NumberCalculusAbram Lipman
Exp series.gifCalculusZhuncheng LiA Taylor series or Taylor polynomial is a series expansion of a function used to approximate its value around a certain point.
Fallacious ProofAlgebraUnknownThe erroneous proof claiming that 1=2. Can you spot the error?
Fibonacci NumbersThe spiral curve of the Nautilus sea shell follows the pattern of a spiral drawn in a Fibonacci rectangle, a collection of squares with sides that have the length of Fibonacci numbers .
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