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Sphericons . . . . . .a new solid shape.
Two cones with a twist.
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|In May 1999 C. J. Roberts, who
lives in the
English town of Baldock, and is a reader of Ian Stewart's "Mathematical
Recreations" column in Scientific American, wrote to Ian about a very
shape that he called a "sphericon." He even enclosed two models
his letter. A mathematical modeler was clearly on the prowl,
for better and more interesting mathematical surfaces.
Steve Mathias, an eternally
living in Sacramento, California, read Stewart's article in the
1999 issue of "SA." He then made several sphericon models.
The NCB now invites you to enjoy slide shows featuring Steve's
Sphericons will make a nice science or math fair display.
Begin with two cones.
the cross-section so as to produce a square. Use the trig
for a 45o - 45o - 90o triangle where
diameter of the cone becomes the hypotenuse
of a plane triangle.
one of the sections.
Reattach the two
The sphericon is a double-cone, two identical cones joined
base to base,
but with a twist. It is easy to make. If you slice a double-cone along
a plane that includes both vertices, you get a rhombic cross section, a
parallelogram with four equal sides. If you use cones of just the right
shape, you get a square cross section. Unlike all other rhombi, the
has an extra symmetry: rotate it through a right angle and it fits back
into the same shape. So you can slice such a double-cone down the
rotate one of the halves through a right angle and glue the two pieces
back together. This is the sphericon. Thanks to the twist, it is not a
double-cone but a much more interesting beast.
The sphericon can also be made from a single piece of thin
to a shape made from four identical sectors of a circle joined together
so that they face in alternate directions (see illustration).
A very delightful property, or characteristic, of the
sphericon is that
it rolls! A cone placed on a flat surface rolls around in circles. A
can roll in a clockwise circle or a counterclockwise one, but it rolls
straight only if you rapidly bowl it or set it on rails. A sphericon
a controlled wiggle, which on average is straight. First one conical
is in contact with the flat surface, then the next. So as it moves
it wiggles alternately to the left and right. It is especially
to start it at the top of a gradual slope and watch it wobble its way
It has one continuous face, and one sphericon can roll around another
If two sphericons are placed next to each other, they can roll
other's surfaces. Four sphericons arranged in a square block can all
around one another simultaneously. And eight sphericons can fit on the
surface of one sphericon so that any one of the outer solids can roll
the surface of the central one.
One reason why the sphericon has such neat geometric
properties is that
its four "edges" - the lines where the component sectors meet - lie
four of the edges of a regular octahedron. The other four edges of the
octahedron correspond to lines that bisect the vertex angles of the
The octahedron, in turn, is closely related to the cube: if you put a
in the middle of each face of a cube and join the dots by straight
you get an octahedron. And cubes, of course, stack in a regular manner
to form a flat layer or fill three-dimensional space.
How do you make a sphericon?
Click here for an enlarged image
suitable for printing
Steve Mathias has been experimenting with the shape and its variations. He has made several sphericons, large and small, out of various materials. He has used wire frames, and dipped them in bubble solution, with surprising and amazing results. He has also made some that exhibit the characteristic wobbly roll, but are dramatically different in appearance.
Recall the study of cones dates to Apollnius (262-190 BC) and other early Greeks.