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

**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.

**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.

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

**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.

**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.

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

**Field**: Algebra**Details:** The Monty Hall problem is a probability puzzle based on the 1960's game show Let's Make a Deal.

**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
.

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

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

**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.

**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.

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

**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.

**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).

**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.

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

**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.

**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.

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

**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.

**Field**: Geometry**Details:**

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

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

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

**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.

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

**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.

**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.

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

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

**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.

**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.

**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.

**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?

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

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

**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.

**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.

**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.

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

**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.

**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.

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

**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.

**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.

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

**Field**: Algebra**Details:** The Monty Hall problem is a probability puzzle based on the 1960's game show Let's Make a Deal.

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

**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.

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

**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.

**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).

**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.

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

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

**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.

**Field**: Geometry**Details:**

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

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

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

**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.

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

**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.

**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.

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

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

**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.

**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?

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

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

**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.

**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.

**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.