Trefoil Klein Bottle

The “Klein Bottle” is what happens when you merge two “Möbius Strips” together: the resulting shape will still have only one side – with its inside and outside merging into one. Obectively, such a paradoxical shape is clearly not possible within our 3-D reality and requires a fourth dimensional jump at some point to make it all come together. Also, because true Klein bottles do not have discernible “inside” or “outside”, they have ZERO VOLUME. As a result, these objects can only be simulated as an “impossible art” in our world, or only modeled with a “fake” 3-D intersection, instead of a true extra-dimensional joint. There are a lot of Klein Bottle model variants, this one is the most intriguing.

Trefoil klein bottle
Triple Klein bottle

Hydrostatic Solution Of Particular Trinominial Equations

A. Demanet devised an interesting method of solution of trinomial equations which depends on the use of communicating vessels of convenient forms.
To solve an equation of the third degree of the form:
x3 + x = c
where c is a constant, an inverted cone and a cylinder, joined together by means of a tube, are taken.
As shown below.

The radius r of the cone and its height h are in the ratio:
r/h = √3/√π
while the base of the cylinder is taken as 1 cm2

Continue reading “Hydrostatic Solution Of Particular Trinominial Equations”

Cube in a Cube or the Intersecting Tetrahedra

A polyhedron compound of two cubes is obtained by allowing two cubes to share opposite polyhedron vertices, and then rotating one a sixth of a turn about the axis that joins the two opposite vertices (see fig. 1 below).

As you can see from fig. 2, the two-cube compound is made up of 12 pyramidal modules. Each pyramidal module is composed of two right triangles with ratio 2:1 and one isosceles right triangle.

Cube in Cube

Print the PDF with the paper model (shown in fig. 3) to make your own compound of two cubes. Continue reading “Cube in a Cube or the Intersecting Tetrahedra”