
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.
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Apollonian Snowflake 
Fractals 
Me (Victor) 
This is a combination of the Apollonian Gasket and the Koch Snowflake, both of which are fractals. The result will be an endless fractal made from two existing fractals. Its really a halfApollonian Gasket because I'm only iterating the largest inscribed circle for each triangle. To start, a construction of the Koch Snowflake must be made because this will be the layout for the circles. On Geometers Sketchpad (GSP, a very useful program that I highly recommend to those who read this page) I am able to make a Koch Curve tool. This tool will let me apply the Koch Curve to any line segment. (Look at the bottom for specific instructions). Simultaneously, I add inscribed circles to my Curve tool. These circles will be my Apollonian aspect of my image. When I increase the iterations of the snowflake, the circles will iterate as well. As cool as this sounds already, adding colors will really make the snowflake ten times more awesome. 
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 
This image shows a pattern of music rhythms generated by Euclidean algorithm. To find out the process of generating music rhythms or how it sounds like, go to section Euclidean Rhythms. 
Arbelos 
Geometry 
csosborne 
This modern knife in the shape of an arbelos is used to make shoes. 
Arbitrage 
Other 
psdGraphics 
Arbitrage is the possibility of making a riskfree profit without investing capital or, alternatively, as riskless instantaneous profit. For example, if one investor could purchase 10 dollars for 9 euros at one bank and then go to a different bank and sell the 10 dollars for 10 euros, he or she would have made a riskless profit of 1 euro and arbitrage would have been achieved. One can see the benefits of arbitrage; it is essentially the process of making free money! However, this reminds one of the adage, “There is no free lunch.” [1] Alas, arbitrage is no exception because, in reality, it does not exist. 
Art Gallery Theorem 
Algebra 
Studio Daniel Libeskind 
The zigzag structure of the Berlin Jewish Museum, known as the Libeskind Building, is one example of the many wacky shaped art museums located around the world. The museum's polygonal shape lends itself to an interesting problem of guard security. 
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 
Algebra 

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 
Geometry 
Wolfram MathWorld 

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 
A Cantor set is a simple fractal that laid the foundation for modern topology. The picture at right is an artistic representation of the Cantor set. 
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 
Algebra 
Phoebe Jiang 
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.
 This is a very famous grid problem in combinatorics, which could be solved by Catalan numbers.

Catenary 
Geometry 
Mtpaley 
A catenary is the curve created by a theoretical representation of a hanging chain or cable held at both ends. 
Change Of Coordinate Transformations 
Other 
Apple Inc. 
An example of various coordinate transformations applied to simple geometry. 
Change of Coordinate Systems 
Calculus 
Brendan John 
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 
Geometry 
Laurens 
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 <b>Euler Spiral</b>. 
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 
The crosscapped disk is one 3 dimensional model of the Real Projective Plane. The crosscapped disk is a 2 dimensional surface that is nonorientable and has only one side. The Real Projective Plane is best represented using 4 spacial dimensions, rather than 3. 
Dandelin Sphere Theory 
Geometry 
Hollister (Hop) David 
This image shows a cone floating on the ocean. a ball floats in the cone with a touch of the ocean surface. A round fish is kissing the ocean surface in the cone. The cone cuts the ocean surface with a "Conic Section", which in the image is an ellipse. 
Dandelin Spheres Theory 
Geometry 
Hollister (Hop) David 
This image shows a head floating on the ocean surface with a funny coneshaped hat. A round fish is kissing the ocean surface under the water in the hat. The hat intersects the ocean surface in a "Conic Section", which in the image is an ellipse. This image is an example of Dandelin Spheres structure. 
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. 
Differentiability 
Algebra 
Lizah Masis 

Dihedral Groups 
Algebra 
3LIAN.COM 
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 
Calculus 
Brendan John 
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 
This image shows the five Platonic solids in the first row, their duals directly below them in the second row, and the compounds of the Platonic solids and their duals in the third row. 
Envelope 
Geometry 
skylighter.com 
<br /><br />
 This is a beautiful blueaerialshell 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.

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. 
Eternal Knot 
Geometry 
Only One Vision Inc. 
The Carrick Mat is a decorative knot, known as the 818 knot in knot theory. 
Euclidean Algorithm 
Number Theory 
Phoebe Jiang 
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 
Algebra 
Unknown 
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 
This is an example of a fractal image called Ford Circles which is a special case of the Apollonian gasket 
Four Color Theorem 
Graph Theory 
Brendan John 
This image shows a four coloring and graph representation of the United States. 
Four Color Theorem Applied to 3D Objects 
Graph Theory 

This picture shows an example of the extension of the four color theorem to nonflat surfaces using a bumpy 3D shape. 
Fourier Transform 
Algebra 

A Fourier Transform changes a function's domain from time to frequency 
Frabjous 
Geometry 
George W. Hart 
Frabjous is a sculpture created by George W. Hart from laser cut aspen wood. The sculpture is constructed from elongated scurve pieces that, when fitted together, create a swirling vortex. 
Fractal Bog 
Fractals 
JeanFrancois Colonna 
This image was obtained by means of a selftransformation of a fractal process. 
Fractal Scene I 
Fractals 
Anne M. Burns 
"Fractal Scene I" is one of Burns' "Mathscapes" and was created using a variety of mathematical forumluas, including fractal methods to generate the clouds and plant life and vector techniques for the colors. 
Fractals With Stars 
Graph Theory 
MWillis 
The final star creation. 
Fun Topology 
Topology 
Paul Nylander 
The topology is equivilent to a sphere with 30 holes. The boundary of each hole loops over itself twice with two ReidemeisterI twists and links with 6 others. 
Gaussian Pyramid 


A Gaussian pyramid is a set of images that are successively blured and subsampled repeatedly. The recursive operation is applied on each step so many levels can be created. Gaussian Pyramids have many computer vision applications, and are used in many places. 
Gradients and Directional Derivatives 
Calculus 
Golden Software 
This image shows gradient vectors at different points on a contour map. These vectors show the paths of steepest descent at different points on the landscape. 
Graph Theory 
Graph Theory 
Awjin Ahn 
This is a graph with six vertices and every pair of vertices connected by an edge. It is known as the complete graph K<sub>6</sub>. 
Graphics Primitives 
Algebra 
Steve Cunningham 
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Hamiltonian Path 
Graph Theory 
Jorin Schug 
In Graph Theory, a Hamiltonian path is a series of vertices and edges such that every vertex is included only once. 
Harmonic Warping 
Calculus 
Paul Cockshott 
This image is a tiling based on harmonic warping operations. These operations take a source image and compress it to show the infinite tiling of the source image within a finite space. 
Harmonies 
Other 

A pianist playing a chord, displaying the harmonies that the multiple notes create 
HarterHeighway Dragon 
Dynamic Systems 
SolKoll 
This image is an artistic rendering of the HarterHeighway Curve (also called the Dragon Curve), which is a fractal. It is often referred to as the Jurassic Park Curve because it garnered popularity after being drawn and alluded to in the novel Jurassic Park by Michael Crichton (1990). 
Henon Attractor 
Dynamic Systems 
Piecewise Affine Dynamics 
This image is a Henon Attractor (named after astronomer and mathematician Michel Henon), which is a fractal in the division of the chaotic strange attractor. 
Hippopede of Proclus 
Topology 
Adam Coffman 
Consider a torus, T, as a surface of revolution, generated by a circle with radius r > 0, and with center at distance R > 0 from the axis... 
Hyperbolic Geometry 
Geometry 
Radmila Sazdanovic 
This is an animation of a square rotating in hyperbolic geometry as represented by the Poincaré Disk Model. 
Hyperbolic Paraboloid 
Calculus 
Unknown 
A hyperbolic paraboloid... 
Hyperbolic Tilings 


This image is a hyperbolic tiling made from alternating two shapes: heptagons and triangles. 
Hyperboloid 
Calculus 
Paul Nylander 
A hyperboloid is a quadric, a type of surface in three dimensions. 
Hypercube 
Geometry 
John Baez 
This is an example of a figure that exists in the 4th dimension. It is the dual to the tesseract. It is also the four dimensional figure that is analogous to the three dimensional octahedron. 
Hypotrochoid 
Geometry 
Victor Luaña 
Three Hypotrochoid curves combined, each represented by a different color: green, yellow, and orange. 
Image Convolution 
Other 

Image Convolution is the process of applying a filter to images. Clockwise from top left, this images shows an original image, a Gaussian Blur filter, a Poster Edges filter, and a Sharpen filter. The filters were applied in Photoshop. 
Implicit Surfaces 
Other 
©Disney Enterprises Inc 
The image to the right is of the character “Flubber” from the 1997 Disney movie of the same title. 
Impossible Geometry 
Geometry 
Lizah Masis 
This image was created by the artist M. C. Escher 
Indra 432 
Other 
Jos Leys 
A Kleinian group floating on the water. 
Inscribed figures 
Geometry 


Inside the Flat (Euclidean) Dodecahedron 
Geometry 
Paul Nylander 
Here is a dodecahedron viewed from the inside with flat mirrored walls. 
Inversion 


This image is an example of a fractal pattern that can be created with repeated inversion in circles. 
Involute 
Geometry 
Chengying Wang 
An involute of a circle can be obtained by rolling a line around the circle in a special way. 
Involute of a Circle 
Geometry 
Wyatt S.C. 
The involute of a circle is a curve formed by an imaginary string attached at fix point pulled taut either unwinding or winding around a circle. 
IsoAxis 
Geometry 
J.A Gutierrez 
A dynamical system that shows the ideal motion of the model is not possible without a slight deformation. 
Iterated Functions 
Algebra 
Anna 

Julia Set 2 
Fractals 
Anna 
This is a filled Julia Set created with a program described in this page. 
Julia Sets 
Fractals 
Anna 
This is a filled Julia Set created with a program described in this page. 
KeplerPoinsot Solids 
Geometry 
Magnus J. Wenniger 
The KeplerPoinsot solids, or polyhedra, are four concave polyhedrons constructed of regular concave polygons. Along with the Platonic Solids, they are referred to as the "cosmic figures". 
Klein Bottle 
Geometry 
3DXM Consortium 
The Klein Bottle is a nonorientable surface with no boundary first described in 1882 by the German mathematician Felix Klein. 
Kleinian Quasifuchsian Limit Set 
Fractals 
Paul Nylander 
Here is a Sunset Moth “blown about” inside a Quasifuchsian limit set. Originally, Felix Klein described these fractals as “utterly unimaginable”, but today we can visualize these fractals with computers. 
Koch's Snowflake 2 
Fractals 
SolKoll 
The image is an example of a Koch Snowflake, which is made by the infinite iteration of the Koch curve. 
Koch Snowflake 
Fractals 
SolKoll 
The image is an example of a Koch Snowflake, a fractal that first appeared in a paper by Swede Niels Fabian Helge von Koch in 1904. It is made by the infinite iteration of the Koch curve. 
Kruskal's Algorithm 
Graph Theory 
Nordhr 
Kruskal’s Algorithm finds a minimum spanning tree in a connected graph with edge weights. 
Kummer Quartic 
Algebra 
3DXM Consortium 
A Kummer surface is any one of a one parameter family of algebraic surfaces defined by a specific polynomial equation of degree four. 
Law of Sines 
Geometry 
Richard Scott 
The law of sines is a tool commonly used to help solve arbitrary triangles. It is a formula that relates the sine of a given angle to its opposite side length. 
Law of cosines 
Geometry 
rscott3 
The law of cosines is a trigonometric generalization of the Pythagorean Theorem. 
Lissajous Curve 
Geometry 
Michael Trott 
<br />This is a beautiful Lissajous Box. The curves on its sides are Lissajous Curves with a frequency ratio of 10:7.<br /><br /> 
Logarithmic Scale and the Slide Rule 
Algebra 
IBM 
This was a picture of an IBM advertisement back in 1953. 
Logarithmic Spirals 
Geometry 
Unknown 
Logarithmic spirals are spirals which appear in nature, such as in this nautilus shell. They possess the remarkable property that the distances between the turnings are in a geometric progression. 
Logistic Bifurcation 
Dynamic Systems 
Diana Patton 
This is a section of a bifurcation diagram. It shows the relationship between a population's potential for growth and its size over time. 
Lorenz Attractor 
Dynamic Systems 
Aaron A. Aaronson 
The Lorenz Attractor is a 3dimensional fractal structure generated by a set of 3 ordinary differential equations. 
Lévy's Ccurve 
Fractals 
SolKoll 
The Lévy's Ccurve is a selfsimilar fractal. 
MILS 04B 
Number Theory 

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.

MILS 04B hlv1 
Number Theory 

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.

MILS 04B hlv2 
Number Theory 

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.

MILS 04B hlv3 
Number Theory 

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.

MILS 04B hlv4 
Number Theory 

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.

MILS 04B hlv5 
Number Theory 

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.

MILS 05 
Number Theory 

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.

Mandelbrot Set 1 
Fractals 
António Miguel de Campos 
An example of a Mandelbrot set. The spiral appears to continue infinitely with each iteration. The spiral will get more detailed the more the viewer zooms in, until the viewer appears to be seeing what he or she began with. 
MarkusLyapunov Fractals 
Dynamic Systems 
BernardH 
MarkusLyapunov fractals are representations of the regions of chaos and stability over the space of two population growth rates. 
Mateko 
Fractals 
Dan Kuzmenka 
Mateko uses different color palettes than image designer Dan Kuzmenka's usual earth tones. He uses fractals to express a spiral without showing the same shape over again. 
Mathematics in architecture 
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. It is an example of how architects use mathematics in design to make the seemingly unbuildable, buildable. 
Mathematics of Gothic and Baroque Architecture 
Geometry 
Blog 
La Sagrada Família (Holy Family) is a Gothic cathedral in Barcelona, Spain designed by Spanish architect Antoni Gaudí. 
Metaballs 
Calculus 

Metaballs are a visualization of a level set of an ndimensional function 
Mobius Strip 
Topology 
David Benbennick 
A Mobius strip, also referred to as a Mobius band, is a bounded surface with only one side and one edge. 
Monkey Saddle 
Calculus 
Mathematica 
This image shows a surface known as a monkey saddle. 
Newton's Basin 
Fractals 
Ashley T. 
Newton's Basin is a visual representation of Newton's Method, which is a procedure for estimating the root of a function. 
Pappus Chain 
Geometry 
Phoebe Jiang 
Pappus chain consists of all the black circles in the pink region. 
Parabola 
Geometry 
Unkown 
A parabola is a ushaped curve that arises not only in the field of mathematics, but also in many other fields such as physics and engineering. 
Parabolic Bridges 
Algebra 
Aaron Logan 
Parabolas are very wellknown and are seen frequently in the field of mathematics. Their applications are varied and are apparent in our every day lives. For example, the main image on the right is of the Golden Gate Bridge in San Francisco, California. It has main suspension cables in the shape of a parabola. 
Parabolic Reflector 
Geometry 
Energy Information Administration 
Solar Dishes such as the one shown use a parabolic shape to focus the incoming light into a single collector. 
Parametric Equations 
Algebra 
Direct Imaging 
The Butterfly Curve is one of many beautiful images generated using parametric equations. 
Parametrization of lines, surfaces and solids 
Geometry 
Matlab, Graphing Calculator 

Pascal's Triangle2 
Algebra 
The Math Forum @ Drexel 
Pascal's Triangle 
Pascal's triangle 
Algebra 
Chengying Wang and the Math Forum at Drexel 

Pentagonal Fractal 
Geometry 
Erin Denenberg, Melina Nolas & Anea Moore 
This is the pentagonal fractal. It incorporates regular decagons, isosceles triangles and regular pentagons. 
Perko pair knots 
Topology 
Diana Patton 
This is a picture of the Perko pair knots. They were first thought to be separate knots, but in 1974 it was proved that they were actually the same knot. 
Permutation 
Algebra 
Photoshop 
The image is a tree of permutations which shows all possible orderings for four colors. 
Permutations 
Algebra 
Photoshop 
The image is a tree of permutations which shows all possible orderings for four colors. 
Pigeonhole Principle 
Other 
mathilluminated 
A pigeon is looking for a spot in the grid, but each box or pigeonhole is occupied. Where should the poor pigeon on the outside go? No matter which box he chooses, he must share with another pigeon. Therefore, if we want all of the pigeons to fit into the grid, there is definitely a pigeonhole that contains more than one pigeon. This concept is commonly known as the pigeonhole principle. The pigeonhole principle itself may seem simple but it is a powerful tool in mathematics. 
Platonic Solid 
Polyhedra 
Abram 
The platonic solids are five regular polyhedra that have faces constructed of congruent convex regular polygons. 
Polar Equations 
Algebra 
chanj 

PopUp Fractals 
Fractals 
Alex and Gabrielle 
This popup object is not just a regular popup—it is also a fractal! 
Pretzel Surface 
Algebra 
3DXM Consortium 
The Pretzel surface is an algebraic surface. 
Prime Numbers in Linear Patterns 
Number Theory 
Iris Yoon 
Create a table with 180 columns and write down positive integers from 1 in increasing order from left to right, top to bottom. When we mark the prime numbers on this table, we obtain the linear pattern as shown in the figure. 
Prime spiral (Ulam spiral) 
Number Theory 
en.wikipedia 

Problem of Apollonius 
Geometry 
Paul Nylander 
This an example of a fractal that can be created by repeatedly solving the Problem of Apollonius. 
Procedural Image 
Computer Graphics 

A procedural image is an image generated by a series of mathematical functions 
Projection of a Torus 
Algebra 
Thomas F. Banchoff 
A torus in four dimensions projected into threedimensional space. 
Pythagorean Tree 
Geometry 
Enri Kina and John Wallison 
A Pythagorean Tree is a fractal that is created out of squares. Starting from an initial square, two additional smaller squares are added to one side of the first square such that the space between all three squares is a right triangle. The side of the larger square becomes the hypotenuse of that right triangle. 
Quadratic Functions in Landmarks 
Algebra 
Teacher's Network 
The Harbour Bridge in Sydney, Australia. The bridge is in the shape of a parabola. 
Quaternion 
Geometry 

Quaternions are a number system that work as an extension of complex numbers by having three imaginary components 
Quipu 
Other 

This is a picture of a quipu (or khipu), a recordkeeping tool used by the Incas. 
Real Projective Plane 
Topology 

This is Boy's surface, one model of the Real Projective Plane in 3 dimensional space. 
Regular Hexagon to Rectangle 
Geometry 

You can use the apothem and perimeter of a regular polygon to find its area. 
Regular Octagon to Rectangle 
Geometry 
Emma F. 
A regular polygon can be "unrolled" to form a rectangle with twice the area of the original polygon. 
Resonance 
Dynamic Systems 
Jeffrey Disharoon 
A picture of a clarinet, an instrument that utilizes a vibrating reed and a resonating chamber to produce sounds. 
Riemann Sphere 
Algebra 
Unknown 

Riemann Sums 
Calculus 
Marhot 
A Riemann sum is an approximation of the area under a curve using a number of rectangles. 
Romanesco Broccoli 
Geometry 
KatoAndLali 
This is the Romanesco Broccoli, which is a natural vegetable that grows in accordance to the Fibonacci Sequence, is a fractal, and is three dimensional. 
Romanesco broccoli 
Fractals 
Jon Sullivan 
Fractals appear in nature, and the Romanesco broccoli is a particularly obvious instance. Along with the fern, the surface of the Romanesco broccoli appears to arise from a fractal reiterated many times. 
Rope around the Earth 
Geometry 
Harrison Tasoff 
This is a puzzle about by how much a rope tied taut around the equator must be lengthened so that there is a one foot gap at all points between the rope and the Earth if the rope is made to hover. Although finding the answer requires only basic geometry, even professional mathematicians find the answer strangely counterintuitive. There is a related problem about stretching the rope taut again where the answer is even more surprising. A question similar to the first appeared in William Whiston's The Elements of Euclid circa 1702. 
Roulette 
Geometry 
Wolfram MathWorld 
Four different roulettes formed by rolling four different shapes and tracing a fixed point on each of these shapes. 
Russell's Antinomy 
Algebra 
Peter Weck 
The blob on the right represents the set of all sets which are not elements of themselves. At first such a set might seem logically acceptable, but it leads straight to a famous contradiction known as Russell’s Antinomy or Russell’s Paradox. 
Set Theory 
Algebra 
Chengying Wang 
At right is an example of a Venn diagram. <br /> Some questions you might ask: What is contained in each colored area? What does the sign "∩" mean? What is a Venn diagram and what is it used for? Set theory can answer these questions. 
Seven Bridges of Königsberg 
Graph Theory 
Bogdan Giu?c? 
The Seven Bridges of Königsberg is a historical problem that illustrates the foundations of Graph Theory 
Siefert surface I 
Algebra 
Jos Leys 
A Seifert surface, a subset of dynamic systems. 
Sierpinski's Triangle 
Geometry 
Unknown 
Sierpinski's triangle is a simple fractal created by repeatedly removing smaller triangles from the original shape. 
Signal Distortion 
Dynamic Systems 
Tim Patterson 2009 
A tube amplifier built with the vacuum tubes intentionally exposed. 
Silhouette Edges 
Geometry 
Steve Cunningham 
This bunny is made up of a group a faces that are adjacent to one another. A program has been run on the object that has found the silhouette edges and they are highlighted in green. This is done by finding which faces have their normals facing towards versus away from the viewer. 
Skull 
Fractals 
Jos Leys 
An abstract skull created by a variation on a fractal colored to achieve the desired image. 
Snell's Law 
Geometry 

This is a picture of a spoon in a glass of water that seems to be bent. Snell's Law is a mathematical formula that predicts the amount of bend seen in the image. 
Social Networks 
Graph Theory 
unknown 
Friend network of a particular Facebook account. The pink indicates a "mob" of tightly interconnected friends, such as high school or college friends. 
Solving Triangles 
Geometry 
Orion Pictures 
In the 1991 film Shadows and Fog, the eerie shadow of a largerthanlife figure appears against the wall as the shady figure lurks around the corner. How tall is the ominous character really? Filmmakers use the geometry of shadows and triangles to make this special effect.
 The shadow problem is a standard type of problem for teaching trigonometry and the geometry of triangles. In the standard shadow problem, several elements of a triangle will be given. The process by which the rest of the elements are found is referred to as solving a triangle.

Sphere Inversion 1 
Geometry 
Jos Leys 
A 3D inversion of a sphere. 
Spiral Explorations 
Geometry 

The picture on the right demonstrates a way of constructing the Fibonacci Spiral (explained below) using arcs. If you are not familiar with the fibonacci sequence, you can click here. Instead of using the squares to determine arcs, this uses circles to determine the arcs (their position and size) that make up the spiral. The traditional squares are overlaid to demonstrate the pattern: the ratio between circle sizes matches the ratio between square sizes. 
Standing Waves 
Dynamic Systems 
Tyler Sammann 
This image depicts a steel string acoustic guitar fret board. This is an instrument which uses standing waves in the strings to produce sounds. 
Steiner's Chain 
Geometry 
fdecomite 
In the image on the right, the Steiner chain consists of a sphere inside another, with a ringlike region in between. This space contains spheres of different diameters but each is tangent to the previous and succeeding spheres as well as to the two nonintersecting spheres. 
Stereographic Projection 
Geometry 
Paul Nylander 
Stereographic projection maps each point on a sphere onto a plane. 
Straight Line and its construction 
Geometry 
Cornell University Libraries and the Cornell College of Engineering 

Strange plant 1 
Fractals 
Jos Leys 
A fractal that looks organic in origin, much like a fern or other plant. Fractals reiterate infinitely, and real ferns seem to grow in the same sort of iterative pattern. 
String Art Calculus 
Calculus 
Diana Patton 
String art is a graphic art form with its roots in Mary Everest Boole's "curvestitching." It became popular as a mode of visual expression in the 1970's, when artists began to use it to create increasingly complex figures. The basis of all string art, though, is one of the main ideas in calculus: the use of straight lines to represent curves. 
Surface Normals 
Geometry 
Nordhr 

Systems of Linear Differential Equations 
Calculus 

Differential equations have always been a popular research topic due to their various applications. A system of linear differential equations is no exception; it can be used to model arms races, simple predator prey models, and more. 
Taylor Series 
Algebra 
Peng Zhao 
Taylor series and Taylor polynomials allow us to approximate functions that are otherwise difficult to calculate. The image at the right, for example, shows how successive Taylor polynomials come to better approximate the function sin(x). In this page, we will focus on how such approximations might be obtained as well as how the error of such approximations might be bounded. 
Tessellations 
Geometry 
Tessellations.org 

Tesseract 
Geometry 
Jason Hise 
The animation shows a threedimensional projection of a rotating tesseract, the fourdimensional equivalent of a cube. 
TestTestTest 
Algebra 
test 
Testing 
Tetra 1 
Geometry 
Jos Leys 
How does one fill a sphere with smaller spheres of various sizes so that every possible void is filled? There are only five known configurations, all obtained by a sphere inversion transformation, the 3D equivalent of a circle inversion. 
The Birthday Problem 
Algebra 
Azavez1 
How many people do you need in a room before it is more than likely that at least two of them have the same birthday? This question is the original Birthday Problem, a common probability problem that stumps many people. In 1970, Johnny Carson tried, and failed, to solve the birthday problem on The Tonight Show.<br />
This page investigates (and solves!) the birthday problem, in addition to some similar puzzles. 
The Fourth Dimension 
Geometry 
Jason Hise 
The fourth dimension refers to the concept of a 4dimensional space, in which four geometric coordinates are necessary to describe any point. In the fourth dimension, our universe is but an infinitesimal slice of the fourth dimension. 
The Golden Ratio 
Algebra 
Joyce Han 

The Logarithms, Its Discovery and Development 
Algebra 
John Napier 

The Monty Hall Problem 
Algebra 
Grand Illusions 
The Monty Hall problem is a probability puzzle based on the 1960's game show Let's Make a Deal.
 When the Monty Hall problem was published in Parade Magazine in 1990, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine claiming the published solution was wrong. It remains one of the most disputed mathematical puzzles of all time.

The Party Problem (Ramsey's Theorem) 
Graph Theory 
Awjin Ahn 
You're going to throw a party, but haven't yet decided whom to invite. How many people do you need to invite to guarantee that at least m people will all know each other, or at least n people will all not know each other? 
The Prisoner's Dilemma 
Algebra 
Greenmantis 
This 2X2 matrix shows the possible actions and resultant outcomes for an instance of the Prisoner's Dilemma. In each outcome box, Robber #1's payoffs are listed to the left, while Robber #2's are on the right. 
The Regular Hendecachoron 
Geometry 
Carlo Sequin 
This object has 11 vertices (shown as spheres), 55 edges (shown as thin cylindrical beams), and 55 triangular faces (shown as cutout frames). Different colors indicate triangles belonging to different cells. 
Three Cottages Problem 
Other 
Unknown 
The three cottage problem is a problem in graph theory. 
Three Dimensional Pythagorean Tree 
Geometry 
Ankur Pawar 

Tone 
Dynamic Systems 
Tyler Sammann 
This image shows the keyboard of a piano, which is a tonal instrument. 
Torus 
Topology 
Lizah Masis 
This picture shows a torus formed by rotating a circle around the zaxis 
Torus Knot 
Geometry 
3DXM Consortium 
In knot theory, a torus knot is a special kind of knot which lies on the surface of an unknotted torus in R3. 
Towers of Hanoi 
Other 
WikiBooks User GeniXPro 

Transformations and Matrices 
Geometry 
Nordhr 
This picture shows an example of four basic transformations (where the original teapot is a red wire frame). On the top left is a translation, which is essentially the teapot being moved. On the top right is a scaling. The teapot has been squished or stretched in each of the three dimensions. On the bottom left is a rotation. In this case the teapot has been rotated around the x axis and the z axis (veritcal). On the bottom right is a shearing, creating a skewed look. 
Tunnel 
Fractals 
Jos Leys 
A fractal image originating from a Mandelbrot set that Jos Leys created using Ultrafractal. 
Vector Fields 
Algebra 
Direct Imaging 
The vector field shown here represents the velocity of a fluid. Each vector represents the fluid's velocity at the point the arrow begins. 
Visualization of Social Networks 
Statistics 
Social Graph 
Friend network of a particular Facebook account. The pink indicates a "mob" of tightly interconnected friends, such as high school or college friends. 
Volume of Revolution 
Calculus 
Nordhr 
This image is a solid of revolution 
Waves 
Algebra 
Xah Lee 

Witch of Agnesi 
Algebra 
John H. Lienhard 

ZSquared Necklace 
Geometry 
Tom Banchoff 
Each subject is the graph of a function of a complex variable, first the complex squaring operation and then the cubing function... 