A closed meander of order 120: a closed curve that intersects a line at 120 points, without intersecting itself.

The Real part of z^z^z

A polyhedron constructed by folding along the edges of a Spidron - a figure in the plane made up of two alternating sequences of isosceles triangles.

A plot of the complex valued function (z-2)²(z+1-2i)(z+2+2i)/z³ by Hans Lundmark

Penrose tiling

Plot of the equation Sin(xSin(y)) = Cos(yCos(x)) by Xah Lee

The scintillating grid illusion

Real part of the j-invariant as a function of the nome q on the unit disk

Mobius strip and knot by Rob Scharein (from The Mobius Strip by Clifford Pickover)

A strange attractor by Nathan Selikoff resembling the Eagle Nebula.

Part of a tessellation of the plane generated using Möbius transforms. By Doug Hensley.

Some examples of Chladni patterns

A rhombic triacontahedron and its net

This is a Klein diagram by Francisco Lara-Dammer that represents A5, the group of symmetries of the icosahedron.

A strange attractor by Nathan Selikoff.

Impossible Geometry by Jos Leys

A pattern showing the result of six levels of recursion using the "inflation rules" of Penrose tiles. By L. Kerry Mitchell.

Kürschák's Dodecagon

A visual proof by Kürschák that a regular 12-sided polygon inscribed in the unit circle has area 3. This means that estimating π to be 3 is like saying a circle is a Dodecagon.

"The Easy Way Up" from Impossible Geometry by Jos Leys

A Mobius-like object with holes by Tom Longtin and Rinus Roelofs.

The interactions between sub-atomic particles at the particle accelerator at CERN in Geneva.

Circle packings

Circle packing is a study of arranging a maximal number of different sized circles (or spheres) so that none overlap. In the picture above, circles of the same size have the same color. See more here.

A beautiful spot in the 3-d Mandelbrot set. To see more check out this gallery.

Conchoids of a 5 cusped epicycloid with pole at center.

Boundary formed by trochoids of different parameters.

Parallels of the Witch of Agnesi by Xah Lee

A fractal dodecahedral subdivision of the sphere

The colors represent the amplitude of the ground's horizontal motion 6 seconds into a 4.4-magnitude earthquake. This image produced on a supercomputer in Pittsburgh is one of several such simulations whose accuracy provides architects with localized information on the kind of intensity buildings may have to withstand.
This is one of many beautiful supercomputer simulations compiled by Discover.

An Apollonian gasket

A cube 6-compound shows 6 overlapping cubes.

A cross-section of a structure generated by n-dimensional cubes by Haresh Lalvani/Patrick Hanrahan with an early 80s computer graphics program. I'd like to see this program adjusted to a modern computer to incorporate color.

This image by Stephen Schiller has been one of my favorites for years. It shows the distribution of fractions of Gaussian integers with a restriction on the denominator. Explicitly, this is the set of complex numbers (a + bi)/(c + di) where a,b,c, and d are integers, and √(c² + d²) < 25. Source: The Pattern Book, by Clifford Pickover.

A Wada Basin is a naturally occurring fractal that can be constructed using 3 mirrored spheres as above.

"This image depicts the interaction of nine plane waves—expanding sets of ripples, like the waves you would see if you simultaneously dropped nine stones into a still pond. The pattern is called a quasicrystal because it has an ordered structure, but the structure never repeats exactly. The waves produced by dropping four or more stones into a pond always form a quasicrystal."

Soap bubbles close up.

Discover's "Most Psychedelic Images in Science"

Geometry of Circles by Phillip Glass

http://demonstrations.wolfram.com/CosineOffsetCurves/

Hyperbolic geometry differs greatly from what we are used to in that the longest path between two points is a straight line. You can think of hyperbolic space like Euclidean space, but with a different way of defining the distance between two points (we call this a metric). Above is the hyperbolic equivalent of an icosidodecahedron. Check out more hyperbolic tilings here.

Clebsch Diagonal Cubic

The surface is given by the equation x^3 + y^3 + z^3 + w^3 + t^3 = 0, restricted to the values x, y, z, w, t, such that x + y + z + w + t = 0.

Above are hypocycloids, curves produced by fixing a point on the circumference of a small circle of radius b, rolling around the inside of a large circle of radius a > b. If a/b is rational, i.e. it can be expressed as a fraction, the path will return to it's starting point. If the ratio is irrational, the path will never touch the same spot on the circumference of the larger circle more than once, and images such as the ones above result.

The first few prime numbers represented in order from left to right in binary.

A mathematical structure known as a Lie algebra, E8 is one of the most complicated Lie algebras, and is used in string theory.

Decagonal trapezohedron

A rhumb line (or loxodrome) runs from the north to the south pole passing through the intersections of the longitude and meridian at the same angle every time.

Einstein Cross: Gravity bends light so that two galaxies look like 5!

Relativity predicts that gravity warps the shape of space and time so that light will no longer move along a straight path. One of the earliest confirmations of the theory was during a total eclipse which allowed us to see a star that wasn't quite in the right place. The sun's gravitational force bent the light rays so that the star appeared about an inch away from where it actually was.
In the image above, the galaxy in the center bends the light waves coming off of the object behind it so that we see 4 different images of the same object. This is called gravitational lensing.

Tensegrity

"Koch Curve 85degrees: Generalizing the von Koch curve with an angle a chosen between 0 and 90°. The fractal dimension is then


http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

Irreducible fractions come up all the time in math problems, but who knew they looked so beautiful all together?
This picture, from Wolfram, is of irreducible fractions in the Complex plane.

These pictures, drawn by a harmonograph, may better be described as pictures of music. A harmonograph is a mechanical device that uses pendulums that move pen and paper simultaneously to create these images. The device was invented by a mathematician in 1844 and apparently was a popular form of entertainment at Victorian soirees.
The images vary depending on how the pendulums swing in relation to one another. Harmonic ratios used in music such as 3:2, 4:3 give the images below.