# HashInfo

***start***

/*/Lightmatter golden gate bridge.jpg*Real Life Parabolas*Aaron Logan*Algebra*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.*Parabolic Bridges<nowiki>

/*/Ghys110.jpg*Seifert surface*Jos Leys*Algebra*A Seifert surface, a subset of dynamic systems.*Siefert surface I<nowiki>

/*/Rose2.gif*A polar rose (Rhodonea Curve)*chanj*Algebra*This polar rose is created with the polar equation: .*Polar Equations<nowiki>

/*/150 extra engineers thumb.jpg*150 __Extra__ Engineers*IBM*Algebra*This was a picture of an IBM advertisement back in 1953.*Logarithmic Scale and the Slide Rule<nowiki>

/*/001.png*Quadratic**Algebra*Just a quadratic function.*Coefficients<nowiki>

/*/Barnsleys-Fern.PNG*Barnsley Fern*Michael Barnsley*Algebra*The Barnsley Fern was created by Michael Barnsley using an iterated function system.*Barnsley Fern<nowiki>

/*/Bezier.png*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.*Bezier Curves<nowiki>

/*/Permutation.jpg*Permutations*Photoshop*Algebra*The image is a tree of permutations which shows all possible orderings for four colors.*Permutations<nowiki>

/*/Kummer.png*Kummer Quartic*3DXM Consortium*Algebra*A Kummer surface is any one of a one parameter family of algebraic surfaces defined by a specific polynomial equation of degree four.*Kummer Quartic<nowiki>

/*/14224 2 peace1.jpg*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. It is an example of how architects use mathematics in design to make the seemingly unbuildable, buildable.*Mathematics in architecture<nowiki>

/*/Napier logtable.jpg*Two Pages from John Napier's Logarithmic Table*John Napier*Algebra*These are two pages from John Napier's original * Mirifici Logarithmorum Cannonis Descriptio (The Description of the Wonderful Canon of Logarithms)* which started with the following

Hic liber est minimus, si spectes verba, sed usum. Sid spectes, Lector, maximus hic liber est. Disce, scies parvo tantum debere libello. Te, quantum magnis mille voluminibus.

which translates into

The use of this book is quite large, my dear friend. No matter how modest it looks, You study it carefully and find that it gives As much as a thousand big books.^{[1]}

*The Logarithms, Its Discovery and Development<nowiki>

/*/OperaHouseandHarbourBridge.jpg*Opera House and Harbour Bridge*Teacher's Network*Algebra*The Harbour Bridge in Sydney, Australia. The bridge is in the shape of a parabola.*Quadratic Functions in Landmarks<nowiki>

/*/ChangeOfBasis.jpg*Change of Basis*Mathematica*Algebra*The same object, here a circle, can be completely different when viewed in other vector spaces.*Basis of Vector Spaces<nowiki>

/*/Screen Shot 2012-04-19 at 10.56.55 AM.png*Controlled 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.*Controlling & Comparing The Blue Wash Fractal<nowiki>

/*/Pretzel.png*Pretzel Surface*3DXM Consortium*Algebra*The Pretzel surface is an algebraic surface.*Pretzel Surface<nowiki>

/*/14224 2 peace1.jpg*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.*Bridge of Peace<nowiki>

/*/Agnesi.jpg*Witch of Agnesi*John H. Lienhard*Algebra*The Witch of Agnesi appears in almost all high school and undergraduate math books. It is a curve that is symmetric about the y-axis and approaches an asymptote on the x-axis. The Witch of Agnesi was originally called La versiera di Agnesi, or "The Curve of Agnesi." "Versiera" is very similar to the word "avversiera" in Italian, which means "woman contrary to God". This was interpreted as witch. The name of the curve was mistranslated.*Witch of Agnesi<nowiki>

/*/prisonerdilemma2.png*The Prisoner's Dilemma*Greenmantis*Algebra*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 Prisoner's Dilemma<nowiki>

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

/*/File:Test.gif*TestTestTest*test*Algebra*Testing*TestTestTest<nowiki>

/*/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<nowiki>

/*/Primitiveobject1.jpg*Graphics Primitives*Steve Cunningham*Algebra*placeholder*Graphics Primitives<nowiki>

/*/Permutation.jpg*Permutation*Photoshop*Algebra*The image is a tree of permutations which shows all possible orderings for four colors.*Permutation<nowiki>

/*/Erroneous proof.png*Fillacious Proof*Unknown*Algebra*The erroneous proof claiming that 1=2. Can you spot the error?*Fallacious Proof<nowiki>

/*/SteamBorrA.jpg*Borromean Rings*Rob Scharein*Algebra*These are Borromean Rings...*Brunnian Links<nowiki>

/*/Russell's Antinomy main.jpg*The Set of All Sets Which Do Not Contain Themselves*Peter Weck*Algebra*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**.*Russell's Antinomy<nowiki>

/*/Riemannsphere1.jpg*Riemann Sphere*Unknown*Algebra*In complex analysis and dynamics, the Riemann Sphere is often used to simplify problems and analysis. It is sometimes denoted as '"`UNIQ--math-00000374-QINU`"', the union of the complex numbers with a point called infinity.*Riemann Sphere<nowiki>

/*/Golden Gate Bridge.jpg*Real Life Parabolas*Aaron Logan*Algebra*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 Integration<nowiki>

/*/k*k**Algebra*k*Circular Rotative Envelope Intersection<nowiki>

/*/Spiral Staircase.jpg*Spiral Staircase**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**.*Cornu Spiral<nowiki>

/*/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<nowiki>

/*/Venn Diagram.jpg*Venn Diagram*Chengying Wang*Algebra*At right is an example of a Venn diagram.

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.*Set Theory<nowiki>

/*/Iterated2.jpg*A function and its first two iterates*Anna*Algebra*This picture is the plot of '"`UNIQ--math-00000373-QINU`"', which is in blue, '"`UNIQ--math-00000374-QINU`"', which is in pink and which we create by plugging in '"`UNIQ--math-00000375-QINU`"' in for x. The yellow line is '"`UNIQ--math-00000376-QINU`"', were we plug in '"`UNIQ--math-00000377-QINU`"' for x.*Iterated Functions<nowiki>

/*/Bounding box.png*Bounding Box*chanj*Algebra*A box bounding the Stanford Bunny mesh.*Bounding Volumes<nowiki>

/*/*e*Abram Lipman*Calculus**Euler's Number<nowiki>

/*/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<nowiki>

/*/hyperboloid.jpg*Hyperboloid*Paul Nylander*Calculus*A hyperboloid is a quadric, a type of surface in three dimensions.*Hyperboloid<nowiki>

/*/x^2_string.gif*String Art*Diana Patton*Calculus*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.*String Art Calculus<nowiki>

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

/*/Mid.gif*Riemann Sums*Marhot*Calculus*A Riemann sum is an approximation of the area under a curve using a number of rectangles.*Riemann Sums<nowiki>

/*/Hyperbolic Paraboloid2.jpeg*Hyperbolic Paraboloid*Unknown*Calculus*A hyperbolic paraboloid...*Hyperbolic Paraboloid<nowiki>

/*/Exponential function Talor Series.gif*Taylor Series*Zhuncheng Li*Calculus*A Taylor series or Taylor polynomial is a series expansion of a function used to approximate its value around a certain point.*Exp series.gif<nowiki>

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

/*/Piano.jpg*Tone*Tyler Sammann*Dynamic Systems*This image shows the keyboard of a piano, which is a tonal instrument.*Tone<nowiki>

/*/Lorenz-attractor-render-1-small.jpg*Lorenz Attractor*Aaron A. Aaronson*Dynamic Systems*The Lorenz Attractor is a 3-dimensional fractal structure generated by a set of 3 ordinary differential equations.*Lorenz Attractor<nowiki>

/*/Tubes.jpg*Signal Distortion*Tim Patterson 2009*Dynamic Systems*A tube amplifier built with the vacuum tubes intentionally exposed.*Signal Distortion<nowiki>

/*/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<nowiki>

/*/Logistic_Bifurcation.gif*Logistic Bifurcation*Diana Patton*Dynamic Systems*This is a section of a bifurcation diagram. It shows the relationship between a population's potential for growth and its size over time.*Logistic Bifurcation<nowiki>

/*/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<nowiki>

/*/Clarinet.jpg*Resonance*Jeffrey Disharoon*Dynamic Systems*A picture of a clarinet, an instrument that utilizes a vibrating reed and a resonating chamber to produce sounds.*Resonance<nowiki>

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

*Anne Burns' Mathscapes<nowiki>

/*/Fractal Broccoli.jpg*Romanesco broccoli*Jon Sullivan*Fractals*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.*Romanesco broccoli<nowiki>

/*/Brokenheart.jpg*Broken Heart*Jos Leys*Fractals*A broken heart created by a variation on a fractal.*Broken Heart<nowiki>

/*/Stage Five.JPG*Pop-Up Fractals*Alex and Gabrielle*Fractals*This pop-up object is not just a regular pop-up—it is also a fractal!*Pop-Up Fractals<nowiki>

/*/Mandelbrot_detail6.jpg*Mandelbrot Set 1*António Miguel de Campos*Fractals*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.*Mandelbrot Set 1<nowiki>

/*/Skull.jpg*Skull*Jos Leys*Fractals*An abstract skull created by a variation on a fractal colored to achieve the desired image.*Skull<nowiki>

/*/Different_Strokes.jpg*Different Strokes*Linda Allison*Fractals*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.*Different Strokes<nowiki>

/*/colorful_circles.png*Apollonian Snowflake*Me (Victor)*Fractals**Apollonian Snowflake<nowiki>

/*/Sol-Koch.jpg*Sol-Koch*SolKoll*Fractals*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.*Koch Snowflake<nowiki>

/*/Light032.jpg*Strange Plant 1*Jos Leys*Fractals*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.*Strange plant 1<nowiki>

/*/Mateko.jpg*Mateko*Dan Kuzmenka*Fractals*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.*Mateko<nowiki>

/*/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<nowiki>

/*/proj_82.jpg*Fractal Bog*Jean-Francois Colonna*Fractals*This image was obtained by means of a self-transformation of a fractal process.*Fractal Bog<nowiki>

/*/FractalSceneI.jpg*Fractal Scene I*Anne M. Burns*Fractals*"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.*Fractal Scene I<nowiki>

/*/Sol-Koch.jpg*Sol-Koch*SolKoll*Fractals*The image is an example of a Koch Snowflake, which is made by the infinite iteration of the Koch curve.*Koch's Snowflake 2<nowiki>

/*/quasifuchsian.jpg*Kleinian Quasifuchsian Limit Set*Paul Nylander*Fractals*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.*Kleinian Quasifuchsian Limit Set<nowiki>

/*/tunnel.jpg*Tunnel*Jos Leys*Fractals*A fractal image originating from a Mandelbrot set that Jos Leys created using Ultrafractal.*Tunnel<nowiki>

/*/Silhouettebunny.png*Silhouette Edges*Steve Cunningham*Geometry**Silhouette Edges<nowiki>

/*/FractalsMandelbrot.jpg*Test Image*Mathematica*Geometry*This is a test image page.*Test<nowiki>

/*/KleinBottle.png*Klein Bottle*3DXM Consortium*Geometry*The Klein Bottle is a non-orientable surface with no boundary first described in 1882 by the German mathematician Felix Klein.*Klein Bottle<nowiki>

/*/Epitrochoid.PNG*Epitrochoids*Albrecht Duerer*Geometry*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.*Epitrochoids<nowiki>

/*/Boy Surface.jpeg*Boy's Surface*Paul Nylander*Geometry*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 3-space was impossible. Today, a large model of Boy's Surface is displayed outside of the Mathematical Research Institute of Oberwolfach in Oberwolfach, Germany. The model was constructed as well as donated by Mercedes-Benz.*Boy's Surface<nowiki>

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

/*/Bouquet1.jpg*Bouquet*George W. Hart*Geometry*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.*Bouquet<nowiki>

/*/escher023a.jpg*Dragons 1*Jos Leys*Geometry*A tessellation created in the style of M.C. Escher.*Dragons 1<nowiki>

/*/Ford.png*Ford Circles*code.haskell.org*Geometry**Ford Circles<nowiki>

/*/Granapic05.jpg*Tile Work of the Lower walls of Salon del Trono, Alhambra, Spain**Geometry*Detailed here is the lower wall of a throne room in the Spanish Muslim palace of the Alhambra. It is considered one of the finest examples of ceramic inlay, radiative geometry, and use of inscribed figures in art.*Inscribed figures<nowiki>

/*/Apothemsgif.gif*What is an Apothem?*azavez1*Geometry*The image to the right shows the shortest distance from the center to the midpoint of one side in various regular polygons.*Apothems and Area<nowiki>

/*/Cardioid_Graph.png*Cardioid*Henrik Wann Jensen*Geometry*A Cardioid is a pattern defined by the path of a point of the circumference of a circle that rotates around another circle.*Cardioid<nowiki>

/*/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<nowiki>

/*/Le pytho.jpg*Pythagorean Tree, in 2 Dimensions*Enri Kina and John Wallison*Geometry*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.*Pythagorean Tree<nowiki>

/*/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<nowiki>

/*/La Sagrada Familia .jpg*La Sagrada Família*Blog*Geometry*La Sagrada Família (Holy Family) is a Gothic cathedral in Barcelona, Spain designed by Spanish architect Antoni Gaudí.*Mathematics of Gothic and Baroque Architecture<nowiki>

/*/*Inside the Flat (Euclidean) Dodecahedron*Paul Nylander*Geometry*Here is a dodecahedron viewed from the inside with flat mirrored walls.*Inside the Flat (Euclidean) Dodecahedron<nowiki>

/*/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<nowiki>

/*/Arbelos(10)-.jpg*Pappus Chain*Phoebe Jiang*Geometry*Pappus chain consists of all the black circles in the pink region.*Pappus Chain<nowiki>

/*/Involute_of_Circle_Animation.gif*Involute*Chengying Wang*Geometry*An involute of a circle can be obtained by rolling a line around the circle in a special way.*Involute<nowiki>

/*/Alhamb.png*Tiling of the Alhambra*Tessellations.org*Geometry*This is a tiling in the Alhambra in Spain, one of the many beautiful designs laid out by the Moors in the 14th Century.*Tessellations<nowiki>

/*/Steiner's_Chain_in_3D.jpg*Steiner's Chain in Third Dimension*fdecomite*Geometry*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.*Steiner's Chain<nowiki>

/*/Image.png*Romanesco Broccoli*KatoAndLali*Geometry*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<nowiki>

/*/Fig-spiro05.png*Hypotrochoid*Victor Luaña*Geometry*Three Hypotrochoid curves combined, each represented by a different color: green, yellow, and orange.*Hypotrochoid<nowiki>

/*/Boy Surface.jpeg*Boy's Surface*Paul Nylander*Geometry*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.*Boy's Surface Vocabulary<nowiki>

/*/normal_Sequin-epostcard.jpg*The Regular Hendecachoron*Carlo Sequin*Geometry*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.*The Regular Hendecachoron<nowiki>

/*/Hexadecachoron.png*Hexadecachoron*John Baez*Geometry**Hypercube<nowiki>

/*/Kepler-Poinsot solids.jpg*Kepler-Poinsot Solids*Magnus J. Wenniger*Geometry**Kepler-Poinsot Solids<nowiki>

/*/Surfacenormalart.png*Surface Normals*Nordhr*Geometry*This picture is of plane segments in different rotations in space. They each have an arrow coming out of their front face at exactly '"`UNIQ--math-00000000-QINU`"' (orthogonal) to the surface.*Surface Normals<nowiki>

/*/NecklaceZ2.jpg*Z-Squared Necklace*Tom Banchoff*Geometry*Each subject is the graph of a function of a complex variable, first the complex squaring operation and then the cubing function...*Z-Squared Necklace<nowiki>

/*/HyGeometryAni_Square.gif*Hyperbolic Geometry*Radmila Sazdanovic*Geometry*This is an animation of a square rotating in hyperbolic geometry as represented by the Poincaré Disk Model.*Hyperbolic Geometry<nowiki>

/*/Ocean.gif*Dandelin Spheres*Hollister (Hop) David*Geometry**Dandelin Spheres Theory<nowiki>

/*/Screen shot 2013-04-30 at 12.29.32 PM.png*Alternative Fibonacci Spiral**Geometry**Spiral Explorations<nowiki>

/*/Escher relativity.jpg*Impossible Geometry of M.C. Escher*Lizah Masis*Geometry*This image was created by the artist M. C. Escher*Impossible Geometry<nowiki>

/*/Tesseract1.gif*The Fourth Dimension*Jason Hise*Geometry**The Fourth Dimension<nowiki>

/*/Frabjous1.jpg*Frabjous*George W. Hart*Geometry*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.*Frabjous<nowiki>

/*/Quaternion.png*Quaternion**Geometry*Quaternions are a number system that work as an extension of complex numbers by having three imaginary components*Quaternion<nowiki>

/*/Hextorec.png*Regular Hexagon to Rectangle with Area of Two Congruent Hexagons**Geometry*You can use the apothem and perimeter of a regular polygon to find its area.*Regular Hexagon to Rectangle<nowiki>

/*/Rope around earth 3.jpg*Rope around the Earth*Harrison Tasoff*Geometry*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.*Rope around the Earth<nowiki>

/*/Tetra1.jpg*Tetra 1*Jos Leys*Geometry*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.*Tetra 1<nowiki>

/*/DualsPlatonicSolids 1000.gif*Duals of Platonic Solids*MathWorld*Geometry**Dual Polyhedron<nowiki>

/*/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<nowiki>

/*/Screen shot 2013-06-11 at 2.22.20 PM.png*Tangents of Crop Circles*Eiman Eltigani*Geometry*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.*Crop Circles<nowiki>

/*/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<nowiki>

/*/Pythagorean_tree.jpg*3-D Pythagorean Tree*Ankur Pawar*Geometry*The **three-dimensional Pythagorean tree** is a fractal composed of cubes whose side lengths are structured according to the Pythagorean theorem. It is called a "tree" because, after a few iterations, the fractal acquires a tree-like shape.*Three Dimensional Pythagorean Tree<nowiki>

/*/Torus_knot.png*Torus Knot*3DXM Consortium*Geometry*In knot theory, a torus knot is a special kind of knot which lies on the surface of an unknotted torus in R3.*Torus Knot<nowiki>

/*/Demo.jpg*Surface Plot of '"`UNIQ--math-00000373-QINU`"'*Matlab, Graphing Calculator*Geometry*This is the Surface Plot of '"`UNIQ--math-00000374-QINU`"'. However, this is not a parametrization. Instead, this is the explicit representation of the surface, that is expressing one '"`UNIQ--balloon-00000375-QINU`"'

in terms of two other variables. A loose parametrization of the surface will be '"`UNIQ--math-00000376-QINU`"'. This is however is not a parametrization in the strictest sense. A parameter is a variable that does not appear in the new expression. Therefore, a few of the more appropriate ones will be '"`UNIQ--math-00000377-QINU`"' or '"`UNIQ--math-00000378-QINU`"' or '"`UNIQ--math-00000379-QINU`"'.*Parametrization of lines, surfaces and solids<nowiki>/*/Octtorec.png*Regular Octagon to Rectangle*Emma F.*Geometry*A regular polygon can be "unrolled" to form a rectangle with twice the area of the original polygon.*Regular Octagon to Rectangle<nowiki>

/*/Involute_of_a_circle.gif*Involute of a Circle*Wyatt S.C.*Geometry*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.*Involute of a Circle<nowiki>

/*/Bolinv001a.jpg*Sphere Inversion 1*Jos Leys*Geometry*A 3D inversion of a sphere.*Sphere Inversion 1<nowiki>

/*/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<nowiki>

/*/Konigsberg bridges.png*Seven Bridges of Königsberg*Bogdan Giuşcă*Graph Theory**Seven Bridges of Königsberg<nowiki>

/*/*Fractals with stars*MWillis*Graph Theory*The final star creation.*Fractals With Stars<nowiki>

/*/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<nowiki>

/*/Hamiltonian_graph1.gif*Hamiltonian Path*Jorin Schug*Graph Theory**Hamiltonian Path<nowiki>

/*/Kruskal's Algorithm.gif*Kruskal’s Algorithm*Nordhr*Graph Theory*Kruskal’s Algorithm finds a minimum spanning tree in a connected graph with edge weights.*Kruskal's Algorithm<nowiki>

/*/Cover_Picture.jpg*Four Color Theorem**Graph Theory*This picture shows an example of the extension of the four color theorem to non-flat surfaces using a bumpy 3D shape.*Four Color Theorem Applied to 3D Objects<nowiki>

/*/PartyProbB.gif*The Party Problem*Awjin Ahn*Graph Theory*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 Party Problem (Ramsey's Theorem)<nowiki>

/*/GraphTheory6.png*Graph*Awjin Ahn*Graph Theory*This is a graph with six vertices and every pair of vertices connected by an edge. It is known as the complete graph *K*_{6}.*Graph Theory<nowiki>

/*/Chryzode.gif*Chryzode*J-F. Collonna &. J-P Bourguigno*Number Theory*Chryzodes are visualizations of arithmetic using chords in a circle.*Chryzodes<nowiki>

/*/Ulam_spiral.png*Ulam Spiral*en.wikipedia*Number Theory*The Ulam spiral, or prime spiral, is a plot in which prime numbers are marked among positive integers that are arranged in a counterclockwise spiral. The prime numbers show a pattern of diagonal lines.*Prime spiral (Ulam spiral)<nowiki>

/*/Screen shot 2012-10-01 at 2.49.15 PM.png*Prime numbers in a table with 180 columns*Iris Yoon*Number Theory*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 Numbers in Linear Patterns<nowiki>

/*/Main_Image-3.jpg*Euclidean Rhythms*Wouter Hisschemöller*Number Theory**Application of the Euclidean Algorithm<nowiki>

/*/Harmonies.jpg*Harmonies**Other*A pianist playing a chord, displaying the harmonies that the multiple notes create*Harmonies<nowiki>

/*/Indra432.jpg*Indra 432*Jos Leys*Other*A Kleinian group floating on the water.*Indra 432<nowiki>

/*/Gold Dollar Sign.jpg*Arbitrage*psdGraphics*Other**Arbitrage<nowiki>

/*/ThreeCottage.jpg*Three Cottage Problem*Unknown*Other*The three cottage problem is a problem in graph theory.*Three Cottages Problem<nowiki>

/*/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<nowiki>

/*/Flubber.jpg*Flubber*©Disney Enterprises Inc*Other*The image to the right is of the character “Flubber” from the 1997 Disney movie of the same title.*Implicit Surfaces<nowiki>

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

/*/*Facebook friend Web*Social Graph*Statistics*Friend network of a particular Facebook account. The pink indicates a "mob" of tightly interconnected friends, such as high school or college friends.*Visualization of Social Networks<nowiki>

/*/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<nowiki>

/*/mainpic134.jpg*Brouwer Fixed Point Theorem*Rebecca*Topology**Brouwer Fixed Point Theorem<nowiki>

/*/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<nowiki>

/*/HippopedeOfProc.gif*Hippopede of Proclus*Adam Coffman*Topology*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...*Hippopede of Proclus<nowiki>

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

/*/Fun Topology.jpg*Fun Topology*Paul Nylander*Topology*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.*Fun Topology<nowiki>

/*/Boy.png*Boy's surface**Topology*This is Boy's surface, one model of the Real Projective Plane in 3 dimensional space.*Real Projective Plane<nowiki>

/*/Vanaelst_thecantorset 2.jpg*Cantor Set*Keith Peters*Topology**Cantor Set<nowiki>

*end

- ↑ The MacTutor History of Mathematics archive, 2006