The Stomachion navigator page is designed to let you tour the enormous possibilities and remarkable properties of stomachions while putting control in your hands. Customize the size and other display features by selecting from the available options, and save the resulting pages and images by copying the links. The Stomachion is a puzzle described by the Greek mathematician Archimedes in the 3rd century BC. It consists of 14 geometric pieces which fit together to form a square. The original texts have survived only fragmentarily and the 17152 solutions to the challenge were discovered only in 2003. Several factors contribute to this great number. There are 3 triangles that only fit in combination with another piece because they share a unique angle, in effect reducing the problem to only 11 pieces. There are 3 pairs of identical triangles that can be considered interchangeable twins, which cuts the number of solutions in half three times. If rotations and reflections are considered identical, the total number of squares is reduced by another factor of eight. This leaves 268 unique stomachions. By exchanging two pieces, or reflecting a distinct (isosceles triangular, rectangular or parallelogramic) segment of the square, one configuration turns into another. In this fashion 266 of the 268 stomachional squares are linked to one another and can be cycled through along what is known as a Hamiltonian path in mathematical jargon. The remaining two squares form a Hamiltonian mini-loop. Display options (key): Pieces (P): draw the reduced 11 or the original 14 pieces. Orientation: rotate and reverse the complete square. Variation: exchange one, two or all of the pairs of twinned triangles. Overlay (V): label the pieces or the complete square. Grid (G): indicate the 12x12 unit matrix of lattice points. Square size (S): shrink or enlarge the square from 16 to 1600 pixels. Colours (C): choose a colour scheme to help identify the pieces. Line width (W): set the thickness of the boundary line between the pieces. Line colour (U): choose the colour of the boundary line between the pieces. Layout (A): choose a collection of related squares, or a number of squares and number of rows. Menu (): hide the navigation from view when not used. Autotour (T): enjoy a slideshow of automatically refreshing pages. Other shortcut keys: B, arrow-left: Previous square N, arrow-right: Next square E: Select by number 1: First square L: Rotate 90° left R: Rotate 90° right D: Rotate 180° M: Mirror F: Flip H: Show help O: Show options
#la the 11 all 14 #or normal rotated right rotated 180° rotated left mirrored right mirrored flipped left mirrored #tw none large medium large 2 small large small small 2 all #po none square name square number capitals by area capitals top down letters by area letters top down Greek capitals Greek top down greek letters greek top down numbers top down numbers bottom up area proportions area ratios angles names custom #grid none points dashes lines #skin blue mono moonlight pastel standard custom #lw 0 1 pixel 2 pixels 3 pixels #cl black blue crimson cyan darkblue darkgreen darkred dimgray dodgerblue firebrick fuchsia gold gray green indigo ivory khaki lightblue lightcyan lime maroon navy none olive orchid pink plum purple red salmon silver slateblue slategray springgreen steelblue tan teal thistle turquoise violet white yellow none custom #m 1 square all rotations all reflections all twins 2 squares 2x2 squares 2x4 squares 12 squares 16 squares 67 squares 134 squares 268 squares custom 1 square #hm always auto no #tick off 1 second 2 seconds 3 seconds 4 seconds 10 seconds 30 seconds