Pages Needing Advanced Explanations

From Math Images

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. Contents 1 The following image pages are in need of image explanations at an advanced level of mathematics (graduate level and beyond). <br style="clear: both" />
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 half-Apollonian 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). Simultaneous, 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	Emma F.	In the image to the right, the apothems in various polygons are shown. If you know the length of the apothem and one side of a regular polygon, you can easily find its area. Explained visually:
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.
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 Fern	Algebra	Sven Geier	The Blue Fern is a fractal, similar to Barnsley's Fern fractal, that was created by Michael Barnsley using an iterated function system.
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 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 Surface	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 three-dimensional space. This object is a single-sided 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 up or to the right in this 8 by 8 grid. 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	J-F. Collonna &. J-P Bourguigno	Chryzodes are visualizations of arithmetic using chords in a circle.
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>.
Cross-cap	Topology	Unknown	The cross-capped disk is one 3 dimensional model of the Real Projective Plane. The cross-capped disk is a 2 dimensional surface that is non-orientable 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 cone-shaped 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	Calculus	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	This is a beautiful blue-aerial-shell firework filling the sky. Each particle of the firework follows a parabolic trajectory, and together they sweep a parabolic area.
Euclidean Algorithm	Number Theory	Phoebe Jiang	This image shows Euclid's method to find the greatest common divisor (gcd) of two integers. The greatest common divisor of two numbers a and b is the largest integer that divides the numbers without a remainder. Here I use 52 and 36 as an example to show you how Euclid found the gcd, so you have a sense of the Euclidean algorithm in advance. As you have probably noticed already, Euclid uses lines, defined as multiples of a common unit length, to represent numbers. First, use the smaller integer of the two, 36, to divide the bigger one, 52. Use the remainder of this division, 16, to divide 36 and you get the remainder 4. Now divide the last divisor, 16, by 4 and you find that they divide exactly. Therefore, 4 is the greatest common divisor. For every two integers, you will get the gcd by repeating the same process until there is no remainder. You may have many questions so far: "What is going on here?" "Are you sure that 4 is the gcd of 52 and 36?" Don't worry. We will talk about them precisely later. This brief explanation is just to preheat your enthusiasm for Euclidean Algorithm! It is amazing to see that he explains and proves his algorithm relying on visual graphs, which is different from how we treat number theory now.
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.
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 is showing a basic understanding of the four color theorem 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 s-curve pieces that, when fitted together, create a swirling vortex.
Fractal Bog	Fractals	Jean-Francois Colonna	This image was obtained by means of a self-transformation 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.
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 Reidemeister-I 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 edges going between each edge to another, also known as the complete graph K_{6}.
Graphics Primitives	Algebra	Steve Cunningham	placeholder
Hamiltonian Path	Graph Theory	Jorin Schug	In Graph Theory, a Hamiltonian path is a series of edges that visits every vertex of a graph exactly 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
Harter-Heighway Dragon	Dynamic Systems	SolKoll	This image is an artistic rendering of the Harter-Heighway 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	Geometry	Jos Leys	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	Geometry	Xah Lee	This image is an example of a fractal pattern that can be created with repeated inversion in circles.
Involute	Geometry	Xah Lee	A colorful illustration of different involutes of a circle obtained by rolling a line around the circle.
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.
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.
Kepler-Poinsot Solids	Geometry	Magnus J. Wenniger	The Kepler-Poinsot 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 non-orientable 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			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.
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 3-dimensional fractal structure generated by a set of 3 ordinary differential equations.
Lévy's C-curve	Fractals	SolKoll	The Lévy's C-curve is a self-similar 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.
Markus-Lyapunov Fractals	Dynamic Systems	BernardH	Markus-Lyapunov 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 n-dimensional 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 u-shaped curve that arises not only in the field of mathematics, but also in many other fields such as physics and engineering.
Parabolic Integration	Algebra	Aaron Logan	Parabolas are very well-known 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. Of course there are many more examples of parabolic architecture such as roller coasters, flight paths, and probably the most recognized, the Golden Arches of McDonald's. With all of these appearances in real life, have you ever wondered how to find the area under one?
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 Triangle	Algebra	The Math Forum @ Drexel	Pascal's Triangle
Pascal's triangle	Algebra	The Math Forum @ Drexel	The first 11 rows of Pascal's triangle are depicted on the right.
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.
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
Pop-Up Fractals	Fractals	Alex and Gabrielle	This pop-up object, made by Alex and Gabrielle, is not just a regular pop-up: it is also a fractal!
Pretzel Surface	Algebra	3DXM Consortium	The Pretzel surface is an algebraic surface.
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 three-dimensional space.
Quaternion
Quipu	Other		This is a picture of a quipu (or khipu), a record-keeping 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
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 counter-intuitive. 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.
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.
Solving Triangles	Geometry	Orion Pictures	In the 1991 film Shadows and Fog, the eerie shadow of a larger-than-life 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.
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 ring-like 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 non-intersecting spheres.
Stereographic Projection	Geometry	Thomas Banchoff	A stereographic projection of 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 "curve-stitching." 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
Taylor Series	Calculus	Unknown	A Taylor series or Taylor polynomial is a series expansion of a function used to approximate its value around a certain point. The animation to the side shows a function and the first eight polynomials. The larger n of the polynomial, the more it looks like the original function.
Tessellations	Geometry	Tessellations.org
Tesseract	Geometry	Jason Hise	The animation shows a three dimensional projection of a rotating tesseract, the four dimensional 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 two of them have the same birthday? This question is the original "Birthday Problem," a common statistics problem that stumps many people. In 1970, Johnny Carson tried, and failed, to solve the birthday problem on The Tonight Show.
The Golden Ratio	Algebra	Azhao1
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 (author)	You're going to throw a party, but haven't yet decided who to invite. How many people do you need to invite to guarantee that at least m people will know each other, or at least n people will not know each other?
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 cut-out frames). Different colors indicate triangles belonging to different cells.
Three Cottages Problem	Other	Unknown	The three cottage problem is a problem in graph theory.
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 z-axis
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	The Towers of Hanoi is a well known puzzle game based on a Hindu legend. According to the story, priests of the Hindu god Brahma were instructed to move 64 golden disks from one of 3 poles to another, and when they completed it the world would end. However, whether the game was inspired by the legend, or the other way around, is unknown.
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.
Volume of Revolution	Calculus	Nordhr	This image is a solid of revolution
Waves	Algebra	Xah Lee
Witch of Agnesi	Algebra	John H. Lienhard
Z-Squared 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...