Close to the infinity
Antonis Margaritis, Andreas Varverakis, Ioanna Mountriza, Alexandros Sygkelakis and Irini Perissinaki
On Wednesday 13th of March 2019, the Medical Museum of the University of Crete organized a day conference with title "Adolesence and brain developement" which was attended by numerous students from many schools (program). Our school, as co-organizer, participated to the activities. The Maths -as the year before- was present with an exhibition about infinity, prepared both by students and teachers. For sure, the infinity is hardly conceived by our brain and gets it to its limits creating paradoxes.
Our exhibits have been categorized in 8 themes -the numbers shown below (which you'll have to click them). Each image can be enlarged by a click on it and by a second click it gets back to the original size.
1. The lemniscate ∞: the symbol of infinity
The symbol of an "eight on its side" is known as the lemniscate and is the symbol of infinity. The English mathematician John Wallis (1616-1703) introduced the symbol to represent mathematical infinity in his Arithmetica Infinitorum of 1655, while the Swiss mathematician Jacob Bernoulli (1654-1705) first called the shape a lemniscus (Latin word for ribbon) in an article in Acta Eruditorum in 1694.
These are our exhibits for the symbol of infinity
The carnival mask
The poster created by Natalia (B3).
The poster was created by Georgia and Elias (B1). See also this article of Math Laboratory.
If we get one photo of the Sun each week the same day and hour (for instance when it gets to its highest point) then in a year we shall have 52 photos which combined will show a solar path in a shape of a non-symmetric lemniscate. The non-symmetry is due to the fact that the Sun moves faster during winter.
The photos below where taken by Anthony Ayiomamiti, here is a link to his collection.
Lemniscate with GeoGebra
It takes only three steps to construct a lemniscate with GeoGebra (download the complete file from here)
You may experiment on the Jacob Bernoullis' lemniscate with this application created by the mathematician Soula Soufari.
The brain lemnisci
When we chose the "infinity" as our subject for the math exhibition in the brain event we had no idea that the brain itself has its own lemnisci...
2. Repetitions: a very simple approach of infinity
In the next image the pots are not infinite. Actually, its just a single pot between two opposite mirrors that reflect one another the image infinite times.
The circular recarsion of images which is continous, gives also a simple impresion of infinity. Students created kaleidocycles and phenakistoscopes like the following.
Phenakistoscopes had been exhibits in our previous year participation to the brain event titled "Brain teasers and optical illusions" (read it here)
3. Infinite sums and paradoxes - part A
Zenon the Eleat (born about at 488 b.C.) devised the paradox of "Achilles and the tortoise", where Achilles couldn't reach a turtle moving ahead in a speed of 1/10 of its own. The paradoxes in the next posters are based on the fact that the result of an infinite sum may be (a) a number, (b) infinity or (c) not existing.
In the first poster created by Georgia and Elias (B1) we can see on the left an optical proof that 1/2+1/4+1/8+...=1.
With a similar argument, the time is not infinite but is actually finite in the case of Achilles and the tortoise.
God has found a way to pay infinite money as the little man has prayed for. It is because 1+1/2+1/3+...=+∞
Something is wrong with the infinite sum (+1)+(-1)+(+1)+(-1)+... at the left side of the poster created by Georgia and Elias (B1). Perhaps this sum does not exist.
4. Infinite sums and paradoxes - part B
The Von Koch snowflake is a fractal with this paradox: While its surface is finite, its perimeter is infinite. This is due to the fact that its construction has infinite stages. In each stage, the perimeter becomes 4/3 greater than what it was in the previous stage.
The construction stages are illustrated in detail in the next image. Click on the button "start" to view them (source)
5. Hilbert's infinite hotel and the many infinities
The German mathematician David Hilbert (23 Jenuary 1862 - 14 February 1943) devised a story about an infinite hotel in order to show the complexity of infinity. You will be surprised with the capability of this hotel:
So, it sounds "outrageous" the set of natural numbers to be equal in number with the set of even numbers. The Set Theory, a branch of Maths, introduces the "cardinal" of a set to count its elements. From this point of view, the set of natural numbers and the set of even numbers have the same cardinal. But this is not the case for the set of real numbers, which has a different cardinal.
In the next poster it is illustrated the "Cantor's diagonal argument" a way to prove that there can not exist a 1-1 and over mapping from the set of natural numbers to the set of real numbers which explains why the two sets have different cardinals.
On the right side of the poster are presented some of the cardinals:
6. The mathematical induction
Natural numbers have got this characteristic: There exists a first (minimal) natural number and for each natural number there exists a succesive one. In this structure is based the "mathematical induction", a proof process which resembles to "domino":
Here there are some statements and their proofs with mathematical induction.
7. Regularities in irrational numbers
The number pi (π) has got infinite decimal digits that do not form some periodicity or any other kind of regularity. This is a characteristic of all irrationals. But when it is written "otherwise" some regularity is evident.
...this "otherwise" refers to continuous fractions. In the right side of the poster there are presented the continuous fractions of the numbers φ (with just units) and π.
In the following pdf file there are more examples.
8. The digits of number π with elastic collisions
Let's get back to the decimal digits of number π. Infinite with no regularities. But there is something interesting on them, as shown in the video:
This site is mastered by Irini Perissinaki iriniper[ατ]sch.gr