Pages Needing Advanced Explanations
The following image pages are in need of image explanations at an advanced level of mathematics (graduate level and beyond).
Field  Author  Description  

Anne Burns' Mathscapes  Fractals  Anne M. Burns  In her Mathscape images, Anne M. Burns combines recursive algorithms for clouds, mountains, and various imaginary plant forms into one picture.

Apollonian Snowflake  Fractals  Me (Victor)  
Apothems and Area  Geometry  azavez1  The 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 Algorithm  Number Theory  Wouter Hisschemöller  
Arbelos  This modern knife in the shape of an arbelos is used to make shoes.  
Arbitrage  Other  psdGraphics  
Barnsley Fern  Algebra  Michael Barnsley  The Barnsley Fern was created by Michael Barnsley using an iterated function system. 
Basis of Vector Spaces  Algebra  Mathematica  The same object, here a circle, can be completely different when viewed in other vector spaces. 
Bedsheet Problem  Take 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 Curves  Algebra  A Bezier Curve involves the use of two anchor points and a number of control points to control the form of a curve.  
Blue Wash  Fractals  Paul Cockshott  This 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 Volumes  Algebra  chanj  A box bounding the Stanford Bunny mesh. 
Bouquet  Geometry  George W. Hart  This is a 9inch diameter tabletop 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 Surface  Geometry  Paul Nylander  Boy'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 3space 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 MercedesBenz. 
Boy's Surface Vocabulary  Geometry  Paul Nylander  While 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 threedimensional space. This object is a singlesided surface with no edges. 
Bridge of Peace  Algebra  The 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 Heart  Fractals  Jos Leys  A broken heart created by a variation on a fractal. 
Brouwer Fixed Point Theorem  Topology  Rebecca  
Brunnian Links  Algebra  Rob Scharein  These are Borromean Rings... 
Buffon's Needle  The 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 Mapping  Algebra  Bump mapping is the process of applying a height map to a lit polygon to give a polygon the perception of depth.  
Cantor Set  Topology  Keith Peters  
Cardioid  Geometry  Henrik Wann Jensen  A Cardioid is a pattern defined by the path of a point of the circumference of a circle that rotates around another circle. 
Catalan Numbers  This 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.
 
Catenary  A catenary is the curve created by a theoretical representation of a hanging chain or cable held at both ends.  
Change Of Coordinate Transformations  An example of various coordinate transformations applied to simple geometry.  
Change of Coordinate Systems  The same object, here a disk, can look completely different depending on which coordinate system is used.  
Chryzodes  Number Theory  JF. Collonna &. JP Bourguigno  Chryzodes are visualizations of arithmetic using chords in a circle. 
Circular Rotative Envelope Intersection  Algebra  k  
Coefficients  Algebra  Just a quadratic function.  
Compass & Straightedge Construction and the Impossible Constructions  Geometry  Wikipedia  This image shows the step by step construction of a hexagon inscribed in the circle using a compass and a unmarked straightedge. 
Conic Section  A conic section is a curve created from the intersection of a plane with a cone.  
Controlling & Comparing The Blue Wash Fractal  Algebra  Different 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 Spiral  Algebra  The 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 Circles  Geometry  Eiman Eltigani  Crop 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. 
Crosscap  Topology  Unknown  
Dandelin Spheres Theory  Geometry  Hollister (Hop) David  
Different Strokes  Fractals  Linda Allison  Different 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 Groups  Each 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 Theorem  The water flowing out of a fountain demonstrates an important theorem for vector fields, the Divergence Theorem.  
Dragons 1  Geometry  Jos Leys  A tessellation created in the style of M.C. Escher. 
Dual Polyhedron  Geometry  MathWorld  
Envelope  
Epitrochoids  Geometry  Albrecht Duerer  An 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 Algorithm  About 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 wellknown 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 Number  Calculus  Abram Lipman  
Exp series.gif  Calculus  Zhuncheng Li  A Taylor series or Taylor polynomial is a series expansion of a function used to approximate its value around a certain point. 
Fallacious Proof  Algebra  Unknown  The erroneous proof claiming that 1=2. Can you spot the error? 
Fibonacci Numbers  The 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 .  
Ford Circles  Geometry  code.haskell.org  
... further results 