This page has proved to be much more popular than I could ever have imagined that it would, and I've heard complaints from several sources that it includes little or no explanation of what the animations illustrate. My original intent was that knowlegeable instructors would use the movies with their students; of course, such instructors would already know what was going on and would provide their own explanations. However, to my amazement and gratification, many more people other than instructors appear to have found something (or things) that they like here--so explanations may now be needed. Over the next few months, I will try to add some.

Bill Emerson, Brad Kline, and I gave an MAA Minicourse in creating and exporting animations like these to the Web at the Joint Mathematics Meetings in New Orleans during January of 2001 and in San Diego during January of 2002. Bill and I also gave a workshop at the Spring 2002 meeting of the Rocky Mountain Section of the MAA in Laramie, WY, in April of 2002. We're always happy to discuss these techniques. Write.

You can save these animations to your hard drive. Performance may improve when you run them from copies stored on your own hard drive. Select "Save As" from your browser's "File" menu. Set the format to "Source" and then save to your hard drive. This will work for machines that don't have QuickTime installed, and you can transfer the resulting .MOV file to another machine that does, using either a floppy or a Zip disk.

At first, I made animations available in a large format (generally 432 pixels square) and a small one (320 pixels square or smaller and--sometimes--with a lower frame rate). So many of the later ones failed to fit into the original square formats, that I've pretty much abandoned the original sizes. But many of the animations appear in a large format and in a small one. The smaller format is for lower-resolution screens and to improve download times.

If you found this page through a web search engine, you may be looking at a copy that your search service cached. In that case, you may not be seeing the page as it currently exists. So you may want to visit Mathematics Animated.

I've received some reports of non-functioning animations. If you encounter one, please let me know.

Permission is granted for non-commercial educational use; all other rights reserved.

The pictures on this page do not move; click on a picture or on one of its associated links in order to access the movies.

A Visual Proof of the Pythagorean Theorem (977K) This is an animated, visual proof of the Pythagorean Theorem. Small format. (306K)

A Visual Proof of Pappus' Generalization of the Pythagorean Theorem (2173K): Let ABC be a triangle, and let parallelograms ABDE and ACFG be erected externally to ABC with respective bases AB and AC. Let H be the point where the sides DE and FG, extended, of these parallelograms meet. If BCJK is the parallelogram erected upon BC, external to triangle ABC and having the sides CJ and BK parallel to and congruent with the segment HA, then the area of BCJK is the sum of the areas of ABDE and ACFG. Small format. (571 K)

The Sine Curve. (396K) Shows how a point moving around a unit circle generates the sine function. Small format. (339K)

The Tangent Curve. (717K) Shows how a point moving around a unit circle generates the tangent function. Small format. (499K)

The Conic Sections. (1785K) This shows how a plane intersects a cone to form the curves traditionally known as the conic sections. It's based on an idea I found on Preston Nichols' web site. (Nichols' version is an animated GIF, instead of QuickTime, and runs only to about 94K.) Small format. (1170K)

An Ellipse. (439K) Using the definition to trace out an ellipse. The horizontal line segment shown below the curve consists of copies of the segments that connect the foci with the moving point on the ellipse, showing that the sum of the two lengths is constant. Small format. (321K)

A proof of Quetelet & Dandelin (2,956K). Sometime around 1825, the Belgian geometers Adolphe Quetelet and Germinal Dandelin devised a simple and elegant construction showing that a plane that is parallel to a generator of the cone intersects that cone in a parabola. (I originally, and erroneously, attributed this proof to Apollonius; I apologize for the error.) Here is an animation of their proof. Small format. (986K)

Another proof of Quetelet & Dandelin (3,572K). The Quetelet/Dandelin proof that a plane whose angle from the vertical is less than the vertex angle of a cone meets that cone in an ellipse. Small format. 1,178K)

An Elliptic Reflector. (371K) A light pulse is emitted from one focus of an elliptical reflector. Small format. (278K)

A Parabolic Reflector. (871K) A planar wave-front enters a parabolic reflector. Small format. (614K)

A Hyperbolic Reflector. (637K) The right half of the hyperbola x^2 - y^2 = 1 is a double-sided reflector here. The left focus emits a pulse of red light. A short time later, the right focus emits a pulse of green light, timed so that the spherical wavefronts resulting from the two pulses both reach the vertex of the reflector simultaneously. Can you predict what happens then? Small format. (464K)

The Moving Secant Line. (184K) Approximating a tangent line by secant lines. Small format. (134K)

Measuring Slope and Plotting the Derivative. (732K) Measures the slopes of lines tangent to a curve and uses the measurements to plot the derivative. Small format. (552K)

The Cycloid. (196K) The curve traced out by a point attached to a circle that rolls along a straight line without slipping. Small format. (168K)

An Epicycloid. (471K) The curve traced out by a point attached to a small circle that rolls without slipping around the outside of a larger circle. The small circle here has radius one-fifth that of the large circle. Small format. (339K)

A Hypocycloid. (492K) The curve traced out by a point attached to a small circle that rolls without slipping around the inside of a larger circle. The small circle here has radius one-fifth that of the large circle. Small format. (357K)

A curve and its length. (984K)

Visualizing the Fundamental Theorem of Calculus. (1.31MB) This movie helps students visualize the area function that lies at the center of the Fundamental Theorem of Calculus.

A Simple Polar Area Problem. (335K) Find the area inside one loop of the curve r = 2 Sin[3t]. Small format. (253K)

A Standard Polar Area Problem. (357K) Find the area inside one loop of the curve r = 2 Sin[3t] but outside of the circle r = 1. Small format. (262K)

A First Look At a Harder Polar Area Problem. (557K) Find the area inside the curve r = 1 + Cos[t] but outside the curve r = Cos[t]. Small format. (404K)

Another Look At a Harder Polar Area Problem. (553K) The same problem as the previous one, but this time showing how the inner area is swept out. Small format. (400K)

A Simple Volume of Revolution About The X-Axis. (316K) The volume of revolution generated by the region between the x-axis and the curve y = Sqrt[x], with x between 1 and 4. Small format not yet available.

Another Simple Volume Of Revolution About The X-Axis. (308K) The volume of revolution that results when the region between the x-axis and the curve y = 2/(1 + (x - 2)^2) is revolved about the x-axis. Small format. (219K)

A Detailed Look at a Volume of Revolution About the X-Axis (4559K) The volume of revolution that results when the region between the x-axis, x = 0, x = 2 Pi, and the curve y = 2 + Sin[x] is revolved about the x-axis. Shows how the surface is generated by revolution and then how disks generate the resulting volume. Small format (1156K)

A Simple Volume of Revolution About the Y-Axis. (531K) The volume of revolution generated by the region between the x-axis and the curve y = Sqrt[x], with x between 1 and 4. Small format. (326K)

Integrating Over a Region in the Plane: Along the x-axis. (1.51 MB) How should we set up an integral over a plane region?

Integrating Over a Region in the Plane: Along the y-axis. (1.84 MB) How should we set up an integral over a plane region?

The Moving Triplet. (2862K) The moving triplet made up of the unit tangent vector, the unit normal vector, and the unit binormal vector for the curve x[t] = (2 + cos[1.5 t]) cos[t]; y[t] = (2 + cos[1.5 t]) sin[t]; z[t] = sin[1.5 t]. Small format. (800K)

How to Make a Contour Map (4320K) Shows how contour maps reflect the surfaces from which they come. Small format. (1140K)

A Singular Surface in Three Dimensions. (833K) The surface z = x y/(x^2 + y^2) has a singularity at x = 0, y = 0. This is a standard example from multivariable calculus. This picture shows a wire-frame skelton embedded in the surface. Small format. (499K)

A Non-Differentiable Surface. (1005K) The surface f[x, y] = Sqrt[Abs[x y]] is continuous at the origin and both of the first order partial derivatives are defined there. Nevertheless, the function is not differentiable there. Embedded wire-frame skeleton. Small Format (654K)

A Singular Surface in Three Dimensions. (560K) The surface z = 5 x^2 y/(x^4 + y^2) has a very interesting singularity at x = 0, y = 0. The limit of z as (x, y) -> (0, 0) along lines of the form y = m x is always zero. Nevertheless, the limit of z as (x, y) -> (0, 0) does not exist; to see this consider what happens to z as (x, y) -> (0, 0) along parabolae of the form y = m x^2. If we define z[0, 0] = 0, this is also an example of a function which is discontinuous at the origin but for which the directional derivatives at the origin exist in every direction. Embedded wire-frame skeleton. Small format. (361K)

Another look of the singular surface in shown above. (711K) This picture shows the same surface, but without the wire-frame skelton and in different lighting. Small format. (444K)

A simple surface pictured with a tangent plane. (8.25 MB) This one needs more work; there're more things I want to show--but it's already too big.

A vibrating drumhead. (1446K) The vibrations of a drumhead after a single, centrally placed hit with a drumstick. The vertical scale is greatly exaggerated so that the effects of higher modes of vibration are discernable. A more complete visual analysis of the vibrating circular membrane is available; it includes a longer version of this animation, as well as others. It also includes depictions of over two dozen of the normal modes of vibration. Write for details. No small format available for this one.

Another vibrating drumhead. (1434K) The vibrations of a drumhead after a single hit, offset from the center with a drumstick. The vertical scale is greatly exaggerated so that the effects of higher modes of vibration are discernable. A more complete visual analysis of the vibrating circular membrane is available; it includes a longer version of this animation, as well as others. It also includes depictions of over two dozen of the normal modes of vibration. Write for details. No small format available for this one.

Inversion In The Complex Plane. (105K) Illustrates the action of the reciprocal function, f[z] = 1/z, in the complex plane.

More Inversion In The Complex Plane. (205K) Another look at the action of the reciprocal function in the complex plane.

Still more Inversion In The Complex Plane. (6073KB) A third look at the action of the reciprocal function in the complex plane.

Squaring In The Complex Plane. (667K) Illustrates the action of the squaring function in the complex plane. Small format. (416K)