There is an ingenious way to take an equilateral triangle, cut it into 4 pieces, and then reassemble the pieces to form a square. This dissection curiosity, described in Howard Eves' A Survey of Geometry, Volume One, is said to have been discovered by Henry Dudeney, who lived in the late19th and early 20th century. On this page, the process for dissection to form a rectangle is shown.

The sketch at right shows the cut up triangle.

Use this page as a base from which to explore various aspects of this dissection: How the triangle is cut up, how to reassemble by using rotations, and how a tiling of the plane by equilateral triangles can be turned into a tiling by rectangles. You may be surprised, as I was, by the amazing variety of patterns that are formed by the animation turning the one tiling into the other.

Below are the links to follow. After going to a link, return to this page by using your browser's Back button, so that you can go on to the next link. At the links the points in red are "live" and can be dragged (try circular motions) to control the animation.

Click HERE for the method of constructing the dissection of a single equilateral triangle.
Reassembling the pieces of a single triangle (shown above) to form a square:
	The first rotation. Use the live point O to put the object into its initial position as a triangle; then drag O counterclockwise 180 degrees to perform the first step in the transformation. The first step transforms the equilateral triangle into a 7-sided polygon, a heptagon.
	Now we have the second rotation. The shape's initial position is the same as the ending position after the first 180 degree rotation. Rotate the live red point to perform the second rotation. (You can click on the animation button to see the path of the red point that you want to drag.) After the second rotation, the heptagon has been transformed into a quadrilateral having two right angles.
	Finally, the last rotation, which completes the transformation of the equilateral triangle into a square.
	Here are all three rotations, plus the translation used to complete the new tiling by squares. The animation is currently running backwards due to some technical difficulties, but you can click the animation button to locate the control point, stop the animation, and then just drag the control point.
Transforming the tiling of the plane by equilateral triangles into a tiling by squares:
	This animation has the tiling of plane by equilateral triangles, with an animation of the first rotation.    -------->
	The animation here is designed to follow the one above. Two of the three pieces rotated above are rotated in this animation.    -------->
	The animation here is designed to follow the two animations above. Only the triangular one of the two pieces rotated immediately above is rotated in this animation.    -------->   At the end of this stage, there is an overlap of the pieces, so that there is no longer a tiling, but ...
	The final animation corrects the overlap by a translation of alternating rows of the blue squares. This completes the transformation of the tiling by equilateral triangles into a tiling by squares. (Except that vertices of squares meet non-vertices.)