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/*/4dtorus.jpg*4-Dimensional Torus*Thomas F. Banchoff*Algebra*A torus in four dimensions projected into three-dimensional space.*Projection of a Torus

/*/Bump-map-demo-bumpy.png*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.*Bump Mapping

/*/Sinusoid implicit flame.png*Sinusoidal waves*Xah Lee*Algebra**Waves

/*/BlueFern.jpg*Blue Fern*Sven Geier*Algebra*The Blue Fern is a fractal, similar to Barnsley's Fern fractal, that was created by Michael Barnsley using an iterated function system.*Blue Fern

/*/17.jpg*Dihedral Symmetry of Order 12*3LIAN.COM*Algebra*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.*Dihedral Groups

/*/Mainimage.jpg*The Monty Hall Problem*Grand Illusions*Algebra*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 Monty Hall Problem

/*/Rose2.gif*A polar rose (Rhodonea Curve)*chanj*Algebra**Polar Equations

/*/NAUTILUS.jpg*Fibonacci numbers in a sea shell*Unknown*Algebra*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.*Fibonacci Numbers

/*/Butterfly1.gif*Butterfly Curve*Direct Imaging*Algebra*The Butterfly Curve is one of many beautiful images generated using parametric equations.*Parametric Equations

/*/180px-Pentagram-phi.svg.png*The Golden Ratio*Azhao1*Algebra**The Golden Ratio

/*/DI_vecfield.jpg*Vector Field of a Fluid*Direct Imaging*Algebra*The vector field shown here represents the velocity of a fluid. Each vector represents the fluid's velocity at the point the arrow begins.*Vector Fields

/*/Coordchange.JPG*Change of Coordinates*Brendan John*Calculus*The same object, here a disk, can look completely different depending on which coordinate system is used.*Change of Coordinate Systems

/*/Metaball.0008.jpg*Metaballs**Calculus*Metaballs are a visualization of a level set of an n-dimensional function*Metaballs

/*/Harmonic warp.jpg*Harmonic Warping of Blue Wash*Paul Cockshott*Calculus*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.*Harmonic Warping

/*/Monkey-Saddle.jpg*Monkey Saddle*Mathematica*Calculus*This image shows a surface known as a monkey saddle.*Monkey Saddle

/*/Fountainflux.gif*Fountain Flux*Brendan John*Calculus*The water flowing out of a fountain demonstrates an important theorem for vector fields, the Divergence Theorem.*Divergence Theorem

/*/Revolutionvolume1.gif*Solid of revolution*Nordhr*Calculus*This image is a solid of revolution*Volume of Revolution

/*/Vase2.gif*Procedural Image**Computer Graphics*A procedural image is an image generated by a series of mathematical functions*Procedural Image

/*/Markus-Lyapunov1.gif*Markus-Lyapunov Fractal*BernardH*Dynamic Systems*Markus-Lyapunov fractals are representations of the regions of chaos and stability over the space of two population growth rates.*Markus-Lyapunov Fractals

/*/DragonCurve.jpg*Harter-Heighway Dragon Curve*SolKoll*Dynamic Systems*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).*Harter-Heighway Dragon

/*/guitarblue.jpg*Standing Waves*Tyler Sammann*Dynamic Systems*This image depicts a steel string acoustic guitar fret board. This is an instrument which uses standing waves in the strings to produce sounds.*Standing Waves

/*/HenonMain.jpg*Henon Attractor*Piecewise Affine Dynamics*Dynamic Systems*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.*Henon Attractor

/*/fs_64_100.gif*Blue Wash*Paul Cockshott*Fractals*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.*Blue Wash

/*/NewtonBasin2.jpg*Newton's Basin*Ashley T.*Fractals*Newton's Basin is a visual representation of Newton's Method, which is a procedure for estimating the root of a function.*Newton's Basin

/*/Anna1.jpg*A Julia Set*Anna*Fractals*This is a filled Julia Set created with a program described in this page.*Julia Sets

/*/Lawofsines copy.jpg*Law of Sines*Richard Scott*Geometry*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 Sines

/*/Shadows and fog.jpg*The Shadow Problem*Orion Pictures*Geometry*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.*Solving Triangles

/*/HyperbolicTiling.jpg*Hyperbolic Tilings*Jos Leys*Geometry*This image is a hyperbolic tiling made from alternating two shapes: heptagons and triangles.*Hyperbolic Tilings

/*/shoemakers Knife.jpg*Arbelos*csosborne*Geometry*This modern knife in the shape of an arbelos is used to make shoes.*Arbelos

/*/Roulette.jpg*Roulette*Wolfram MathWorld*Geometry*Four different roulettes formed by rolling four different shapes and tracing a fixed point on each of these shapes.*Roulette

/*/BuffonNeedle 700.gif*Buffon's Needle*Wolfram MathWorld*Geometry**Buffon's Needle

/*/SpiderCatenary2.jpg*Catenary*Mtpaley*Geometry*A catenary is the curve created by a theoretical representation of a hanging chain or cable held at both ends.*Catenary

/*/Fountain.jpg*Parabola*Unkown*Geometry*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.*Parabola

/*/Refraction-of-light36.jpg*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.*Snell's Law

/*/S35-1.jpg*Drawing a Straight Line*Cornell University Libraries and the Cornell College of Engineering*Geometry**Straight Line and its construction

/*/Tranformations4.png*Transformations*Nordhr*Geometry*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.*Transformations and Matrices

/*/HexagonConstructionAni.gif*Creating a regular hexagon with a ruler and compass*Wikipedia*Geometry*This image shows the step by step construction of a hexagon inscribed in the circle using a compass and a unmarked straightedge.*Compass & Straightedge Construction and the Impossible Constructions

/*/Sierpinski clear.gif*Sierpinski's triangle*Unknown*Geometry*Sierpinski's triangle is a simple fractal created by repeatedly removing smaller triangles from the original shape.*Sierpinski's Triangle

/*/Tesseract1.gif*Tesseract*Jason Hise*Geometry*The animation shows a three dimensional projection of a rotating tesseract, the four dimensional equivalent of a cube.*Tesseract

/*/Alhamb.png*Tiling of the Alhambra*Tessellations.org*Geometry**Tessellations

/*/Blue-aerial-shell.jpg*Blue-aerial-shell*skylighter.com*Geometry*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.*Envelope

/*/Apollonian.jpg*Apollonian Gasket*Paul Nylander*Geometry*This an example of a fractal that can be created by repeatedly solving the Problem of Apollonius.*Problem of Apollonius

/*/Law_of_cosines_pictiure.jpg*Law of Cosines*rscott3*Geometry*The law of cosines is a trigonometric generalization of the Pythagorean Theorem.*Law of cosines

/*/Solardish.jpg*Parabolic Reflector Dish*Energy Information Administration*Geometry*Solar Dishes such as the one shown use a parabolic shape to focus the incoming light into a single collector.*Parabolic Reflector

/*/Konigsberg bridges.png*Seven Bridges of Königsberg*Bogdan Giu?c?*Graph Theory*The Seven Bridges of Königsberg is a historical problem that illustrates the foundations of Graph Theory*Seven Bridges of Königsberg

/*/Cover_Picture.jpg*Four Color Theorem**Graph Theory*This picture is showing a basic understanding of the four color theorem using a bumpy 3D shape.*Four Color Theorem Applied to 3D Objects

/*/PartyProblemA.gif*The Party Problem*Awjin Ahn (author)*Graph Theory*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 Party Problem (Ramsey's Theorem)

/*/Usagraphfinal2.PNG*Four Color Theorem*Brendan John*Graph Theory*This image shows a four coloring and graph representation of the United States.*Four Color Theorem

/*/EA1.jpg*Euclidean Algorithm*Phoebe Jiang*Number Theory*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.*Euclidean Algorithm

/*/Ulam_spiral.png*Ulam Spiral*en.wikipedia*Number Theory**Prime spiral (Ulam spiral)

/*/ImageConvolution.jpg*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.*Image Convolution

/*/Quipu3.jpeg*Quipu**Other*This is a picture of a quipu (or khipu), a record-keeping tool used by the Incas.*Quipu

/*/Perko knots.gif*Perko pair knots*Diana Patton*Topology*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.*Perko pair knots

/*/CrossCapTwoViews.PNG*Cross-cap and Cross-capped Disk*Unknown*Topology*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.*Cross-cap

/*/mainpic134.jpg*Brouwer Fixed Point Theorem*Rebecca*Topology**Brouwer Fixed Point Theorem

/*/Mobius strip.jpg*Mobius Strip*David Benbennick*Topology*A Mobius strip, also referred to as a Mobius band, is a bounded surface with only one side and one edge.*Mobius Strip *end