# 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.

Field Author Description
Arbitrage Other psdGraphics Arbitrage is the possibility of making a risk-free profit without investing capital or, alternatively, as risk-less 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 risk-less 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.
Basis of Vector Spaces Algebra Mathematica The same object, here a circle, can be completely different when viewed in other vector spaces.
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 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 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 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.
Brunnian Links Algebra Rob Scharein These are Borromean Rings...
Chryzodes Number Theory J-F. Collonna &. J-P Bourguigno Chryzodes are visualizations of arithmetic using chords in a circle.
Coefficients Algebra Just a quadratic function.
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>.
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.
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.
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.
Fractal Bog Fractals Jean-Francois Colonna This image was obtained by means of a self-transformation of a fractal process.
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.
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.
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.
Inside the Flat (Euclidean) Dodecahedron Geometry Paul Nylander Here is a dodecahedron viewed from the inside with flat mirrored walls.
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.
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.

## Contents

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.
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 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.
Pascal's triangle Algebra Chengying Wang and the Math Forum at Drexel
Quadratic Functions in Landmarks Algebra Teacher's Network The Harbour Bridge in Sydney, Australia. The bridge is in the shape of a parabola.
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.
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.
Skull Fractals Jos Leys An abstract skull created by a variation on a fractal colored to achieve the desired image.
Sphere Inversion 1 Geometry Jos Leys A 3D inversion of a sphere.
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.
TestTestTest Algebra test Testing
The Logarithms, Its Discovery and Development Algebra John Napier
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.
Tunnel Fractals Jos Leys A fractal image originating from a Mandelbrot set that Jos Leys created using Ultrafractal.
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.
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...