On this page I introduce and describe special kaleidocycles. I number them with K1 to K15.
K2 The smallest amount is eight tetrahedra.
Making of the Ring You can produce the ring with paper. You can make the eight tetrahedra separately and connect them with adhesive tape. This is simple, but long work. It is more challenging to design a net of the ring, but folding it is a bit more difficult. The length of a triangle is e.g. 2.5 cm; then the drawing fits an A4 page. (2) Cut out the net. (3) Go over the lines with a ballpen or the blunt edge of scissors, so that you can fold the paper at the lines more easily. Fold the paper several times. (4) First form the tetrahedra 1111 and 2222 at the same time. Glue the triangles xx (top) on the triangles 12 (down). (5) Repeat the process with the pairs 3333/4444, 5555/6666 and 7777/8x8x in this order. (6) Close the ring by gluing the triangles 88 on xx last. If you want to make rings with 10,12,14, ... tetrahedra,
put in more strips.
K3
There is a right angled triangle with the sides b, a and a/2. The Pythagorean theorem gives b²=a²+(a/2)² or b=sqrt(5)/2*a. You have 4-sided pyramids instead of tetrahedra.
M.C.Escher Kaleidocycles top
K4
You have 4-sided pyramids instead of tetrahedra. The triangles are no longer equilateral but isosceles.
Template: Origin: Randolf Rehfeld
K5 (2) Reflect a triangle on a side. (3) Erect a rectangle on the hypotenuse of the reflected triangle. The marked lines are equal. (4) Put the triangle on the rectangle. (5) Repeat this process five times. Fold a pyramid of the four yellow right angled triangles. (6) Form a row and add strips on the triangles to fix the pyramids and to close the ring later. Result: You get a kaleidocycle of six pyramids with a
triangular hole.
Template Description
To explain the name "invertible cube", you may think to look at the well known hexagon inside a cube. You can see the triangles inside and outside the hexagon, but I found no relation to the whole pyramids. If you can use the 3D-view, you recognize the six basic sides of the pyramids. This explains the name "invertible cube":
You recognize the six pyramids of the ring. The cube is
not solid. The pyramids only fill a third of the cube. You can prove it:
Calculation
If you want to complete the cube as a solid cube, you need two equal "bolts" to put into the hollow cube. You find a template of this figure on Marcus Engel's homepage.
K6 How to make it (2) Reflect a triangle on a side. (3) Erect a rectangle on the hypotenuse of the reflected triangle. The marked lines are equal. (4) Put the triangle on the rectangle. (5) Repeat this process seven times. Fold a pyramid of the four right triangles. (6) Form a row and add strips on the triangles to fix the pyramids and to close the ring later. Template This ring is called half
octahedron here, because you can fold this ring to a (red) half octahedron.
Origin: Randolf Rehfeld (URL below) - There is a template.
You get more kaleidocycles, if you start with higher regular polygons.
K15
In this partition the cube is called Schneider-Würfel on Margarita Ehrlich's and Ellen Pawlowski's German web sites (URL below).
K7 How to make this ring
Six squares and equilateral triangles on top form the net.
There is a similarity with the invertible cube described above. In both cases there are positions with a triangle as silhouette. But it isn't closed as Schatz cube is. The edge a/2 is slightly shorter than sqrt(3)/3*a=0,56a at the Schatz cube. This ring ought to be a hint to look for new kaleidocycles and to work on them. You could already read above that there is an endless number of kaleidocycles. A "normal" kaleidocycle is always formed if there are two opposite edges in pairs and a common perpendicular orthogonal. (Marcus Engel, kaleidocycles_theory.pdf).
Cube One is a new cube puzzle invented by the graphic designer and artist Dieter A.W. Junker from Kassel / Germany. The challenge is to form a cube of 2x2 Kaleidocycles.
Each kaleidocycle is a puzzle itsself. One is leading to a tetrahedron, the other to an octahedron.
The challenge is to form a tetrahedron with two Kaleidocycles. Both kaleidocycles only differ in the order of the pyramids.
But it is easier to use the templates which are available on the internet. I add the names Ki in my list of links and books, if you can find a template. Have fun in building kaleidocycles and in looking for new shapes.
German Annemarie Honegger
Dieter A. W. Junker
Franz Zahaurek
Gymnasium Hechingen (Mathewerkstatt)
Gymnasium Königsbrunn
Paul Schatz Stiftung
Randolf Rehfeld (Wundersames Sammelsurium)
English Akira Nishihara
Arvind Gupta
Dave Love and Bill Haneberg
David Singmaster
Enchanted Learning online
FoldPlay
G. Korthals Altes
Jaap Scherphuis
Maurice STARCK
Kristina
Burczyk's Kaleidocycles
Marcus Engel
MATHEMATICAL CONCEPTS, inc
Simon Quellen Field (Science Toys)
YouTube
French Hubert Martineau
Peuplier (Le Forum en Papier)
(2) Gerald Jenkins and Anne Wild, Make Shapes 1, Diss
(Norfolk), Tarqin Publications, 1998 K1
This page is also available in German. URL of
my Homepage:
© 2005 Jürgen Köller |