### Bidimensional Müller-Lyer Illusion

I am working on a new two-dimensional variant of the Müller-Lyer illusion… You may be surprised to know that the Müller-Lyer illusion isn’t only linear: it involves plane geometry too! In fig. A shown below, the ends of the blue and red collinear segments, arranged in a radial fashion around a central point, delimit two perfectly concentric circles. However, for most observers, they seem instead to define a large ovoid that circumscribes another one, slightly eccentric (Fig. B). This comes from the fact that the red segments seem to stretch towards the lower part of the figure, while the blue segments seem to stretch towards the upper part of the same. As you can see, in this variant comes also into play the “neon color spreading” effect. In fact, a bluish inner oval-like shape appears within the black arrow heads (Fig. A), though the background is uniformly white.

### Sum of Infinite Power Series

Have a look at the two distinct sums of series of powers below.

Same procedure, different result accuracy levels… Can you guess what went wrong in the operation of fig. 2?

### 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.

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

### 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.

### Numbers In Nature

Diaethria phlogea, the “89’98 butterfly”, is a species of butterfly of the Nymphalidae family found in Colombia, South America. The markings on its wing resemble a painted number: an 89, a 98 or even an 88.
Are there any other animals with numbers painted on their body?

### 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.

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”