# Pages Needing More Mathematical Explanations

[Again, straighten out Image Author vs. Student Author disparity. GK]

### The following image pages are in need of explanations that incorporate more mathematical details and content.

FieldDescription
ArbitrageOther
Basis of Vector SpacesAlgebraThe same object, here a circle, can be completely different when viewed in other vector spaces.
Boy's SurfaceGeometryBoy'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 VocabularyGeometryWhile 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 PeaceAlgebraThe 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.
Buffon's NeedleThe Buffon's Needle problem is a mathematical method of approximating the value of pi $(\pi = 3.1415...)$involving repeatedly dropping needles on a sheet of lined paper and observing how often the needle intersects a line.
ChryzodesNumber TheoryChryzodes are visualizations of arithmetic using chords in a circle.
Cornu SpiralAlgebraThe 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.
Different StrokesFractalsDifferent 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.
Dragons 1GeometryA tessellation created in the style of M.C. Escher.
Dual PolyhedronGeometry
Fractal BogFractalsThis image was obtained by means of a self-transformation of a fractal process.
Gaussian PyramidA 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.
HyperboloidCalculusA hyperboloid is a quadric, a type of surface in three dimensions.
HypercubeGeometry
Impossible GeometryGeometryThis image was created by the artist M. C. Escher
Indra 432OtherA Kleinian group floating on the water.
Inside the Flat (Euclidean) DodecahedronGeometryHere is a dodecahedron viewed from the inside with flat mirrored walls.
Kleinian Quasifuchsian Limit SetFractalsHere 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.
Law of cosinesThe law of cosines is a trigonometric generalization of the Pythagorean Theorem.
MILS 04B hlv2The 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.
MatekoFractalsMateko 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 of Gothic and Baroque ArchitectureGeometryLa Sagrada Família (Holy Family) is a Gothic cathedral in Barcelona, Spain designed by Spanish architect Antoni Gaudí.
Pascal's triangleDepicted on the right are the first 11 rows of Pascal's triangle, one of the best-known integer patterns in the history of mathematics. Each entry in the triangle is the sum of the two numbers above it. Pascal's triangle is named after the French mathematician and philosopher Blaise Pascal (1623-1662), who was the first to write an organized work about the properties and applications of the triangle in his treatise, Traité du triangle arithmétique (Treatise on Arithmetical Triangle) in 1653.<ref name = "wiki:Pascal's triangle">Wikipedia (Pascal's Triangle). (n.d.). Pascal's Triangle. Retrieved from http://en.wikipedia.org/wiki/Pascal%27s_triangle</ref><ref>Pickover, Clifford A.(2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics Sterling. ISBN 978-1-4027-5796-9</ref>
Pascal's triangle plays important roles in probability theory, in algebra, in combinatorics, and in number theory.
Pigeonhole PrincipleA 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.
Quadratic Functions in LandmarksAlgebraThe Harbour Bridge in Sydney, Australia. The bridge is in the shape of a parabola.
QuaternionGeometryQuaternions are a number system that work as an extension of complex numbers by having three imaginary components
Regular Hexagon to RectangleGeometryYou can use the apothem and perimeter of a regular polygon to find its area.
Regular Octagon to RectangleGeometryA regular polygon can be "unrolled" to form a rectangle with twice the area of the original polygon.
ResonanceDynamic SystemsA picture of a clarinet, an instrument that utilizes a vibrating reed and a resonating chamber to produce sounds.
RouletteGeometryFour different roulettes formed by rolling four different shapes and tracing a fixed point on each of these shapes.
Seven Bridges of KönigsbergGraph Theory
Siefert surface IAlgebraA Seifert surface, a subset of dynamic systems.
SkullFractalsAn abstract skull created by a variation on a fractal colored to achieve the desired image.
Sphere Inversion 1GeometryA 3D inversion of a sphere.
Straight Line and its constructionGeometry
Strange plant 1FractalsA 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.
TestTestTestAlgebraTesting
The Golden Ratio
The Logarithms, Its Discovery and DevelopmentAlgebraThese 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.<ref>The MacTutor History of Mathematics archive, 2006</ref>

The Regular HendecachoronGeometryThis 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 ProblemOtherThe three cottage problem is a problem in graph theory.
ToneDynamic SystemsThis image shows the keyboard of a piano, which is a tonal instrument.
TunnelFractalsA fractal image originating from a Mandelbrot set that Jos Leys created using Ultrafractal.
Visualization of Social NetworksStatisticsFriend network of a particular Facebook account. The pink indicates a "mob" of tightly interconnected friends, such as high school or college friends.
Z-Squared NecklaceGeometryEach subject is the graph of a function of a complex variable, first the complex squaring operation and then the cubing function...