How to ‘magically’ untie a shoelace double knot

Topology is a fascinating branch of mathematics that describes the properties of an object that remain unchanged under continuous “smooth” deformations. Actually, many 3D puzzles are based on topological principles and understanding some very basic principles may help you analyze whether a puzzle is possible or not.

Puzzle-Meister G. Sarcone created this amusing everyday-life topological puzzle to help children to easily take their shoes off.

image knots

As you know, the standard shoelace knot is designed for quick release and easily comes untied when either of the working ends is pulled. Thus, most people think that tying a shoelace into a double knot is an effective method of making the knot “permanent”. But is it true? Continue reading “How to ‘magically’ untie a shoelace double knot”

Rowboat Dilemma

A rowboat is floating in a harbor, and a stubborn donkey pulls by mean of a long rope through a pulley the boat toward the shore. When the donkey has moved 1 meter, how far has the boat moved:
a) exactly 1 meter,
b) more than 1 meter,
c) less than 1 meter?


show solution

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”

The Plant That Is Able To Count Almost To Five

We already knew birds can count, but what about plants? Is this idea so surrealist? No, it isn’t because research says the carnivorous plant with a suggestive name, Venus Flytrap (also referred to ‘Dionaea muscipula’), snaps its jaws shut only when the tiny hairs on the surface of the trapping structure formed by two lobes have been stimulated twice within a 20-second window. An additional stimulation primes the trap for digestion. Five stimulations trigger the production of digestive enzymes – and more additional hairs’ stimulations mean more enzymes.

Dionoea muscipula Continue reading “The Plant That Is Able To Count Almost To Five”

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”