All Images

The Prisoner's Dilemma

Field: Algebra
Details: 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.

4-Dimensional Torus

Field: Algebra
Details: A torus in four dimensions projected into three-dimensional space.

Taylor Series

Field: Algebra
Details: Taylor series and Taylor polynomials allow us to approximate functions that are otherwise difficult to calculate. The image at the right, for example, shows how successive Taylor polynomials come to better approximate the function sin(x). In this page, we will focus on how such approximations might be obtained as well as how the error of such approximations might be bounded.

Sinusoidal waves

Field: Algebra

Dihedral Symmetry of Order 12

Field: Algebra
Details: 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.

Barnsley Fern

Field: Algebra
Details: The Barnsley Fern was created by Michael Barnsley using an iterated function system.

A polar rose (Rhodonea Curve)

Field: Algebra

The Monty Hall Problem

Field: Algebra
Details: 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.

Fibonacci numbers in a sea shell

Field: Algebra
Details: 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 .

The Golden Ratio

Field: Algebra

Butterfly Curve

Field: Algebra
Details: The Butterfly Curve is one of many beautiful images generated using parametric equations.

Vector Field of a Fluid

Field: Algebra
Details: The vector field shown here represents the velocity of a fluid. Each vector represents the fluid's velocity at the point the arrow begins.

Solid of revolution

Field: Calculus
Details: This image is a solid of revolution

Change of Coordinates

Field: Calculus
Details: The same object, here a disk, can look completely different depending on which coordinate system is used.

Harmonic Warping of Blue Wash

Field: Calculus
Details: 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.

Fountain Flux

Field: Calculus
Details: The water flowing out of a fountain demonstrates an important theorem for vector fields, the Divergence Theorem.

Markus-Lyapunov Fractal

Field: Dynamic Systems
Details: Markus-Lyapunov fractals are representations of the regions of chaos and stability over the space of two population growth rates.

Standing Waves

Field: Dynamic Systems
Details: This image depicts a steel string acoustic guitar fret board. This is an instrument which uses standing waves in the strings to produce sounds.

Harter-Heighway Dragon Curve

Field: Dynamic Systems
Details: 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

Field: Dynamic Systems
Details: 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.

Sol-Koch

Field: Fractals
Details: 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.

Blue Wash

Field: Fractals
Details: 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.

Newton's Basin

Field: Fractals
Details: Newton's Basin is a visual representation of Newton's Method, which is a procedure for estimating the root of a function.

A Julia Set

Field: Fractals
Details: This is a filled Julia Set created with a program described in this page.

Apollonian Gasket

Field: Geometry
Details: This an example of a fractal that can be created by repeatedly solving the Problem of Apollonius.

Law of Sines

Field: Geometry
Details: 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.

Parabolic Reflector Dish

Field: Geometry
Details: Solar Dishes such as the one shown use a parabolic shape to focus the incoming light into a single collector.

The Shadow Problem

Field: Geometry
Details: 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.

Romanesco Broccoli

Field: Geometry
Details: 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.

Arbelos

Field: Geometry
Details: This modern knife in the shape of an arbelos is used to make shoes.

Roulette

Field: Geometry
Details: Four different roulettes formed by rolling four different shapes and tracing a fixed point on each of these shapes.

Buffon's Needle

Field: Geometry

Catenary

Field: Geometry
Details: A catenary is the curve created by a theoretical representation of a hanging chain or cable held at both ends.

Transformations

Field: Geometry
Details: 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.

Parabola

Field: Geometry
Details: 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.

Drawing a Straight Line

Field: Geometry

Creating a regular hexagon with a ruler and compass

Field: Geometry
Details: This image shows the step by step construction of a hexagon inscribed in the circle using a compass and a unmarked straightedge.

Tesseract

Field: Geometry
Details: The animation shows a three-dimensional projection of a rotating tesseract, the four-dimensional equivalent of a cube.

Tiling of the Alhambra

Field: Geometry

Blue-aerial-shell

Field: Geometry
Details:

This is a beautiful blue-aerial-shell firework filling the sky. Each particle of the firework follows a parabolic trajectory, and together they sweep an area with the red curve as its boundary. This red boundary is then called the envelope of those parabolas. What's more, as we are going to see in the following sections, this envelope also turns out to be a parabola.

Lissajous Box

Field: Geometry
Details:
This is a beautiful Lissajous Box. The curves on its sides are Lissajous Curves with a frequency ratio of 10:7.

Law of Cosines

Field: Geometry
Details: The law of cosines is a trigonometric generalization of the Pythagorean Theorem.

The Party Problem

Field: Graph Theory
Details: 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?

Four Color Theorem

Field: Graph Theory
Details: This image shows a four coloring and graph representation of the United States.

Seven Bridges of Königsberg

Field: Graph Theory
Details: The Seven Bridges of Königsberg is a historical problem that illustrates the foundations of Graph Theory

Euclidean Algorithm

Field: Number Theory
Details: About 2000 years ago, Euclid, one of the greatest mathematician of Greece, devised a fairly simple and efficient algorithm to determine the greatest common divisor of two integers, which is now considered as one of the most efficient and well-known early algorithms in the world. The Euclidean algorithm hasn't changed in 2000 years and has always been the the basis of Euclid's number theory. This image shows Euclid's method to find the greatest common divisor of two integers. The greatest common divisor of two numbers a and b is the largest integer that divides the numbers without a remainder.

Ulam Spiral

Field: Number Theory

Perko pair knots

Field: Topology
Details: 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.

Cross-cap and Cross-capped Disk

Field: Topology
Details: 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.

Brouwer Fixed Point Theorem

Field: Topology

Mobius Strip

Field: Topology
Details: A Mobius strip, also referred to as a Mobius band, is a bounded surface with only one side and one edge.

Algebra

The Prisoner's Dilemma

Field: Algebra
Details: 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.

4-Dimensional Torus

Field: Algebra
Details: A torus in four dimensions projected into three-dimensional space.

Taylor Series

Field: Algebra
Details: Taylor series and Taylor polynomials allow us to approximate functions that are otherwise difficult to calculate. The image at the right, for example, shows how successive Taylor polynomials come to better approximate the function sin(x). In this page, we will focus on how such approximations might be obtained as well as how the error of such approximations might be bounded.

Sinusoidal waves

Field: Algebra

Dihedral Symmetry of Order 12

Field: Algebra
Details: 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.

Barnsley Fern

Field: Algebra
Details: The Barnsley Fern was created by Michael Barnsley using an iterated function system.

A polar rose (Rhodonea Curve)

Field: Algebra

The Monty Hall Problem

Field: Algebra
Details: 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.

Fibonacci numbers in a sea shell

Field: Algebra
Details: 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 .

The Golden Ratio

Field: Algebra

Butterfly Curve

Field: Algebra
Details: The Butterfly Curve is one of many beautiful images generated using parametric equations.

Vector Field of a Fluid

Field: Algebra
Details: The vector field shown here represents the velocity of a fluid. Each vector represents the fluid's velocity at the point the arrow begins.

Calculus

Solid of revolution

Field: Calculus
Details: This image is a solid of revolution

Change of Coordinates

Field: Calculus
Details: The same object, here a disk, can look completely different depending on which coordinate system is used.

Harmonic Warping of Blue Wash

Field: Calculus
Details: 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.

Fountain Flux

Field: Calculus
Details: The water flowing out of a fountain demonstrates an important theorem for vector fields, the Divergence Theorem.

Dynamic Systems

Markus-Lyapunov Fractal

Field: Dynamic Systems
Details: Markus-Lyapunov fractals are representations of the regions of chaos and stability over the space of two population growth rates.

Standing Waves

Field: Dynamic Systems
Details: This image depicts a steel string acoustic guitar fret board. This is an instrument which uses standing waves in the strings to produce sounds.

Harter-Heighway Dragon Curve

Field: Dynamic Systems
Details: 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

Field: Dynamic Systems
Details: 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.

Fractals

Sol-Koch

Field: Fractals
Details: 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.

Blue Wash

Field: Fractals
Details: 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.

Newton's Basin

Field: Fractals
Details: Newton's Basin is a visual representation of Newton's Method, which is a procedure for estimating the root of a function.

A Julia Set

Field: Fractals
Details: This is a filled Julia Set created with a program described in this page.

Geometry

Apollonian Gasket

Field: Geometry
Details: This an example of a fractal that can be created by repeatedly solving the Problem of Apollonius.

Law of Sines

Field: Geometry
Details: 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.

Parabolic Reflector Dish

Field: Geometry
Details: Solar Dishes such as the one shown use a parabolic shape to focus the incoming light into a single collector.

The Shadow Problem

Field: Geometry
Details: 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.

Romanesco Broccoli

Field: Geometry
Details: 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.

Arbelos

Field: Geometry
Details: This modern knife in the shape of an arbelos is used to make shoes.

Roulette

Field: Geometry
Details: Four different roulettes formed by rolling four different shapes and tracing a fixed point on each of these shapes.

Buffon's Needle

Field: Geometry

Catenary

Field: Geometry
Details: A catenary is the curve created by a theoretical representation of a hanging chain or cable held at both ends.

Transformations

Field: Geometry
Details: 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.

Parabola

Field: Geometry
Details: 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.

Drawing a Straight Line

Field: Geometry

Creating a regular hexagon with a ruler and compass

Field: Geometry
Details: This image shows the step by step construction of a hexagon inscribed in the circle using a compass and a unmarked straightedge.

Tesseract

Field: Geometry
Details: The animation shows a three-dimensional projection of a rotating tesseract, the four-dimensional equivalent of a cube.

Tiling of the Alhambra

Field: Geometry

Blue-aerial-shell

Field: Geometry
Details:

This is a beautiful blue-aerial-shell firework filling the sky. Each particle of the firework follows a parabolic trajectory, and together they sweep an area with the red curve as its boundary. This red boundary is then called the envelope of those parabolas. What's more, as we are going to see in the following sections, this envelope also turns out to be a parabola.

Lissajous Box

Field: Geometry
Details:
This is a beautiful Lissajous Box. The curves on its sides are Lissajous Curves with a frequency ratio of 10:7.

Law of Cosines

Field: Geometry
Details: The law of cosines is a trigonometric generalization of the Pythagorean Theorem.

Graph Theory

The Party Problem

Field: Graph Theory
Details: 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?

Four Color Theorem

Field: Graph Theory
Details: This image shows a four coloring and graph representation of the United States.

Seven Bridges of Königsberg

Field: Graph Theory
Details: The Seven Bridges of Königsberg is a historical problem that illustrates the foundations of Graph Theory

Number Theory

Euclidean Algorithm

Field: Number Theory
Details: About 2000 years ago, Euclid, one of the greatest mathematician of Greece, devised a fairly simple and efficient algorithm to determine the greatest common divisor of two integers, which is now considered as one of the most efficient and well-known early algorithms in the world. The Euclidean algorithm hasn't changed in 2000 years and has always been the the basis of Euclid's number theory. This image shows Euclid's method to find the greatest common divisor of two integers. The greatest common divisor of two numbers a and b is the largest integer that divides the numbers without a remainder.

Ulam Spiral

Field: Number Theory

Topology

Perko pair knots

Field: Topology
Details: 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.

Cross-cap and Cross-capped Disk

Field: Topology
Details: 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.

Brouwer Fixed Point Theorem

Field: Topology

Mobius Strip

Field: Topology
Details: A Mobius strip, also referred to as a Mobius band, is a bounded surface with only one side and one edge.

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