A deserving scientist of such a name, especially a mathematician, experiences in his work the same impression as an artist; his pleasure is so big and of the same nature.
Jules Henri Poincaré
INFINITUM. Mathematical appointments
Although many people do not understand it, the mathematician experiences a magnificent sensation realizing his work. As there says Poincaré, a big pleasure; and, as says Isa, a subidón.
What do you think?
I leave the problem of this week to you, in this case related to the year in which we are:
To demonstrate that for any one verifies positive real numbers that
Luck.
This article is a collaboration sent for fede to gaussianos (25 pound) gmail (point) com.
Brief biographical Nagel critique 
Christian Heinrich von Nagel
Christian Heinrich von Nagel, German geometer, was born on February 28, 1803 in Stuttgart, Germany, and German died on October 27, 1882 in also Ulm.In 1821 Nagel began to study Theology, finishing his studies in 1825. But during these four years his interests also went towards the mathematics and the physics.
So much it was so he became a teacher of mathematics of secondary in the German city of Tübingen. But the thing did not stay there. In 1826 Nagel confers a doctorate thanks to his triangulis rectangulis work on ex-algebraic aequatione construendis (On triangles rectangles construibles from an algebraic equation). Later, in 1830, Nagel moves to Ulm where it is employed at the Gymnasium (school of secondary preparatory for top studies) of this locality.
His principal contribution to the mathematics is framed in the geometry of the triangle. In this article we are going to see, between other things, two constructions related to the triangle that take his name: the point of Nagel and the line of Nagel.
Introduction As the distance of the baricentro to an apex is the double of the distance of to the average point of the opposite side, the homotecia with center and reason-1/2 transforms the triangle, antimedial or anticomplementary of, in the triangle, and this one in his medial or complementary triangle.
The applet GeoGebra-Java could not have executed.
In geometry of the triangle it is called sometimes a complement of a point to his image in the homotecia and anticomplement of to his image in the homotecia
The point, a point, his complement, and his anticomplement they are aligned, and placed so that it is the average point of and.
If in the figure we place the point in the circuncentro of, the point is the circuncentro of the antimedial triangle (that is the ortocentro of), the point is the circuncentro of the medial triangle (that is to say the center of the circle of 9 points), and the line is the line of Euler of the triangle.
On the other hand if we place the point in the uncenter of, the point is the uncenter of the antimedial triangle, the point the uncenter of the medial triangle, and the line is the line that in Wolfram MathWorld they have decided to call somewhat arbitrarily a Nagel line, for the fact that the uncenter of the antimedial triangle is the Nagel point, as we will demonstrate next.
On the other hand the Spieker point is for definition the uncenter of the medial triangle, and of the previous remarks one concludes that the uncenter, the baricentro, the point of Spieker and the point of Nagel are aligned, it is the average point of the segment and.
The Nagel point 
We call ceviana of Nagel to the line, in the figure, which joins an apex with the point of touch of the exwritten circumference opposite to the apex with the opposite side.
The point, to the being the intersection of the common tangent one to the circumferences written and exwritten put up to with the line that joins the centers of these circumferences is a center of a homotecia that transforms the circumference exinscribed in the written circumference. This homotecia takes the radio to the radio, parallel to and therefore perpendicularly to.
Therefore the ceviana happens for the point diametrically put up in the circle inscribed to the point of touch of this circle with the side put up to.
Since we saw in the post on the circles tritangentes and therefore if it is the average point of.
Since also, it proves that the parallel lines and they are.
And as the homotecia, which transforms the triangle into his antimedial one, transforms the line into a line that passes for parallel to, that is to say in the Nagel ceviana $AE$, turns out that the Nagel cevianas meet in a point, the point of Nagel and this point is the uncenter of the antimedial triangle.
Of the post on circles tritangentes we conclude also that the Nagel cevianas bisect the perimeter of the triangle, that is to say two parts of the perimeter of the triangle placed to one and another side of every ceviana of Nagel they have equal length.
The Spieker pointThe Spieker point is the center of the circle inscribed in the medial triangle, or I circulate of Spieker, and it has some quite interesting properties.
If in the figure it is the average point of and we prolong the side up to so that, and it is the average point of, and they are parallel and.
Camo $BE$ is perpendicular to $AH$, and this line is the exterior bisectriz of, $A_1F$ it is parallel to the interior bisectriz of, and therefore it is a bisectriz of the medial triangle.
Therefore the lines that join the average point of every side with the Spieker point, that is to say the bisectrices of the medial triangle, bisect the perimeter of the triangle, like the cevianas of Nagel.
If, the segments are respectively equal to the segments, and the average points of these equal segments are placed at the same distance of the straight line.
Then the center of gravity of a mass distributed uniformly by the perimeter of the triangle is in the line $A_1F$. Since also it is in other bisectrices of the medial triangle, it turns out that the Spieker point is the center of gravity of the perimeter of the triangle.
The average point of equidistant es of the points of touch of the exwritten circumferences put up to and with the side, and therefore it is in the radical axis of these circumferences.
As the radical axis is perpendicular to the line that joins the centers, which it is the exterior bisectriz of the angle in, it turns out that the radical axis of two exwritten circumferences is the bisectriz of the medial triangle, and therefore the Spieker point is the radical center of three exwritten circumferences, that is to say the tangent ones from the Spieker point have the same length to the exwritten circumferences.
The circumferences of Jenkins of sound three tangent circumferences internally to an exwritten circumference and outwardly to others two.
Three Jenkins circumferences are cut in the Spieker point, since the investment with regard to the orthogonal circle to three exwritten circumferences, which center is the Spieker point, transforms the sides of the triangle into the Jenkins circumferences.
And also if the Spieker point is on the circumference inscribed in, three circumferences of Jenkins are tangent to a straight line perpendicular to the line of Nagel, and in another case the center of the tangent circumference to three Jenkins circumferences is in the Nagel line, because this circumference is inverse of the written circumference.
Certainly the latter point is not, seems to me, in ETC.: Will it be new? According to Geogebra his first coordinate trilineal for (6,9,13) is 166.495. and it is not on the search page of ETC.
The following figure tries to illustrate the previous properties.
The applet GeoGebra-Java could not have executed.
Sources used for the biographical critique:
In company of friends, the writers can discuss on his books, the economists on the state of the economy, the lawyers his last suits and the businessmen his last procurement, but the mathematicians cannot speak on his mathematics by no means. And the deeper is his work, the less understandable it is.
Alfred Adler
INFINITUM. Mathematical appointments
I agree with Adler. For a mathematician it is very complicated to explain to someone who should not be much put in the matter what is what it does. It is sure that some of you you have been in a situation like that one day. The comments are the best way of counting your experiences.
Torrent Gossip Girl S03E14 The Lady Vanished freeThis article has been promoted to appear in the front of Wiggle me. If you have liked and want to vote for it enter this linkage and bundle click in Wiggle it.
Introduction 
In a letter directed to the Swiss physicist Nicolas Béguelin, Euler he was commenting on the following thing:
All the suppressed numbers of only one form prime in sound or cousins' doubles where and they are prime between themselves. I have observed that other similar expressions of the form enjoy the same property giving to the letter suitable values.
This is, any number that can express itself of the only form how, for and relative cousins, it is prime or the double of a cousin. In particular, any odd number that could express itself of the only form in the previous sense is prime.
But there is still more. Not only it serves an expression of the type, but certain values exist of such that an expression of the type fulfills the same property. To these values of es to which there are called they numeri idonei (suitable numbers or suitable numbers in Spanish and suitable numbers or idoneal numbers in English).
At least this was the initial definition of suitable number. But this way of defining this type of numbers presents some problems. For example, it is a suitable number (we will see it further on) and for him it is fulfilled that:
it is the only representation of the number 9 as. But since we all know 9 it is not prime, although yes it is a potency of a cousin, since. Therefore we should say that it is a suitable number if any odd number that could express itself of the only form as it is prime or promotes of a cousin, but it is possible to play in tune a little more to eliminate this new possibility, this is, that the number is a potency of a prime number (in the first linkage of the sources you can see some of the conditions that we can add to him to the definition to avoid this).
Knowing a little the way of working of any Euler he can imagine that it did not remain there, that his investigations on this topic did not end in the establishment of the definition of this type of numbers. Knowing about his character investigator one he tends to think that it tried to study in depth more the matter. And having little information about his achievements it is not difficult to become convinced of that it did it, and very deeply. Since yes, this way it was. Euler prepared a list of suitable numbers. It is the following one:
In whole 65 numbers that Euler verified that they were suitable (in the sense commented previously). In fact it investigated more: it used it is ready to construct prime numbers even of eight numbers.
Come to this point the most logical thing is that we rise the following question to ourselves: is the set of suitable numbers infinite? The answer is not. In 1934, the mathematician Sarvadaman Chowla demonstrated that the set of suitable numbers is finite.
Knowing this us another question arises: are there more suitable numbers apart from the found ones by Euler? Unfortunately for this question there is no answer yet, although yes information is had. Specifically it is known that like much one more suitable number exists, apart from that they are in the list. And that if the latter suitable number in fact exists, must be major than 100000000.
Major prime number met the suitable numbersWe have mentioned before Euler that it used these numbers to find prime numbers relatively grades (up to eight numbers). The biggest prime number that Euler found with this téctica was. To demonstrate that this number of eight numbers is prime it would be necessary to verify that the only solution of the equation
It already does enough time we comment on a curious property of the number 26. Specifically it is this one:
The number 26 is the only natural number that is placed between a square () and a bucket ().
Apparently it was Fermat who demonstrated the above mentioned result, but in the post where we realized of this characteristic of 26 no test of this fact was happening. It was Juanbuffer who was contributing in a comment a pdf with a demonstration of the same one (that if I do not remember badly was not in Spanish). Unfortunately it seems that it is already not possible to gain access to the above mentioned document (at least I cannot). For this motive I started looking … and I have found it. My admired Carlos Ivorra is who has provided the above mentioned test to me. Very well, in fact I do not know if it is his, but it appears in one of the books in format pdf that it has available in his web: Theory of Numbers.
In this article you are going to be able to see this demonstration.
In fact the demonstration that I am going to present to you of the fact of which the only natural number is 26 with the mentioned property previously is relatively elementary. The interesting of the test is that it leaves of the set of the natural numbers to demonstrate a characteristic in. The fact to rest on a set major that to demonstrate something in him is a quite useful, made argument of which many mathematicians took advantage when they became convinced of the potency of the above mentioned argument.
Centrémonos in the topic. We are going to do the demonstration in (the entire numbers). Then the statement of the result to demonstrate the following one:
Theorem:
The only entire solutions of the equation
they are.
Demonstration:
A simple glance to the equation says to us that it cannot be an even number. If out we would have that also it would be a pair. The contradiction would be in the fact that the right part of the equality would be divisible between 8, but the left part would not be not even divisible between 4. Therefore it has to be an odd number.
We leave now of for penetrating into the ring. We think that the previous equation in this ring his expression can happen factorizada of the following way:
We consider in this ring the following norm:
It is simple to verify that the above mentioned norm is multiplicativa, this is, that it is positive for any element different from zero of, that zero is zero for the element and that the norm of a product of two elements of es the product of the norms of saying elements.
Let's suppose now that they fulfill the initial equation and let's take the elements and of. Any element that is their common divisor two must divide also to his sum, and to his difference. Taking norms in this situation we would have the following thing:
Therefore. The only pairs of values that fulfill this are them following:
With the first two possibilities we obtain the elements and-1$ of, that are units of this ring. In other cases we obtain the elements and, all of them with norm pair (2 ó 4), therefore they cannot divide to, whose norm () is odd.
With this we come to the following thing: and they are prime between themselves.
Now, we had the initial equation factorizada of the following form:
Joining these two facts we have that the product of two elements of which they are prime between themselves is equal to a bucket. It forces that each of these elements is he itself a bucket. In particular:
Let's develop now the right part of the latter equality:
Equaling coefficients of of the initial and final expressions we come to the following equality:
A simple analysis of the values of and it takes us that the only possible values are and (let's remember that and there are entire numbers). For obtenemos that and this implies that. And for obtenemos and therefore, that is the looked result.
Do you know any other demonstration of this fact? The comments are yours.
In view of the title of the post there is quite clear the subject-matter of the problem of this week: truth? There it goes:
He calculates the following highest common factor:
Luck.
It is not possible that numbers lacking in interest exist, then, to they be, the first one of them would be already interesting because of the same absence of interest.
Martin Gardner
INFINITUM. Mathematical appointments
Interesting reasoning of mister Gardner, who also comes to the hair after seeing the interesting property of the number 26 that I showed you does a pair of days.
The university is one of the best stages in the life of a student, at least under my point of view. In this epoch of the academic life one penetrates into a completely new world, in which multitude of histories lives and in the one that meets many people.
At least this was my case. I was lucky to meet very good persons in my university stage in Granada, persons who helped me very much in those moments and with whom I shared unforgettable experiences. Unfortunately they always remain the people to whom you do not go so far as to be related so much, although there is no reason for it. The people who shares every day with you but with whom you do not have so much contact.
Although it has been a few years since it already finished this period I keep on remembering many of my partners, so much to the most nearby (clearly) as those who it were not so much. Isa belongs, unfortunately, to the latter group. And I say unfortunately because she always looked like to me a magnificent person, always with a smile in the mouth, always ready to throw a hand. And, penetrating already into the academic part, because she was always a brilliant student. And when I say brilliant I mean tremendously brilliant. Lola, one of his friends in that epoch (I do not know if before already beginning the career you knew each other), can confirm that Isa was always over all that we share class with her. For this motive it does not surprise me that it has come up to where it has come. And for being as it is an atrocity I am glad.
Elizabeth Fernández
Elizabeth Fernández born in Linares on August 16, 1979, began his Licentiate in Mathematical Sciences in the course 1997-98 in the University of Granada. According to its own words, from the beginning there attracted attention of him the geometry (to none of that we know your sublime Geometry course III with Paco Martín misses this to us). Even such a point came the thing that he enjoyed in his last two years of career of two fellowships in the department of Geometry and Topology in the above mentioned university.After happening for Murcia and Badajoz, at present she is an Employed Teacher She confers a doctorate of the department of Applied Mathematics on I of the University of Seville.
In the direction of the ICM Very well: and what is what Isa has obtained? Since a little so important as to be the first Spanish woman who receives an invitation to help like speaker in the ICM. Almost not at all. 
Pablo Mira
The above mentioned invitation came to him for his works on surfaces of constant average curvature and so much she like his partner Pablo Mira is going to be who communicate the results that they have obtained in this most important congress.The news on this invitation came to me to the ICM across Wiggle me (the linkage is at the end of this article). Nothing more to see it I started looking for a way of contacting with Isa to congratulate it and to mention to him that he wanted to write something on her in Gaussianos, blog that, certainly, already knew across Lola (thank you). Later a few post office crossed, Isa mentioned to me that he would try to write something telling of what there consists the work that Pablo and she they have realized and that has served to them to go to the ICM. Who better than she to explain it?
The works of Isa and PabloVery well, it is already being time for we to know why they have invited the ICM to Isa and Pablo. The following text is what Isa has written for me and for all of you explaining his works to us.
Surfaces of constant average curvature (CMC)
A very important concept at the time of studying surfaces is that of the average curvature, which gives us a measurement of how the surface bows in the space. The idea is the following one: for every point P of the surface we consider all the normal sections of the surface that happen for this point, which there are the curves that are obtained on having cut the surface with all the planes perpendicular to the same one in the point P. Of all these curves we remain with those who have minor and major curvature (the called principal directions), these directions mark the maximum thing that we can bow towards a side or towards other in the surface.
If we call and to the curvatures of two principal directions, the average curvature in this point is, precisely, the arithmetical average between two o'clock:
The surfaces that have the same average curvature in all his points name half constant (CMC) surfaces of curvartura, and have geometric properties that make them very interesting.
For example, the surfaces of equal CMC to zero are named a minimal surfaces, name that comes from the fact of which these surfaces are those who have minor area of between all the surfaces with the same outline (locally, that is to say, considering sufficiently small pieces of the surface). This property is precisely the one that it characterizes to the movies of soap (it is the first one of the famous laws of Plateu, which govern the behavior of the movies of soap). This allows us to characterize the minimal surfaces as those in that, if we cut away a small piece of the surface and put the rest of the surface in water with soap, the movie that forms in the hollow left by the clipping has exactly the same form as the original piece.
Very well, everything previous was referring to surfaces through that they live inside the usual space (the space euclídeo three-dimensional), but the CMC surfaces in general, and the minimal surfaces in particular, exist in any type of spaces, and a branch of the surface Theory of big relevancy at present is the study of the surfaces of CMC in homogeneous spaces. And what is a homogeneous space? Since said to big features, it is a space that is equal in all his points, that is to say, it does not have special points (although yes there can be special directions). Obviously the space euclídeo is a homogeneous space, but there is more. Let's think for example about a cylinder on a sphere, that is to say, the space product. In this cylinder all the points are equal, but it turns out to be clear that the vertical direction (that of the factor) is special. The three-dimensional homogeneous spaces are very studied, and his classification keeps many relation with the famous geometries of Thurston, related to the conjecture of Poincaré.
And this is the field in which Pablo Mira and it was me who have been employed at last years (from 2005) and on that they have invited us to give a conference in the ICM (that probably will take for title Thusrton 3 - dimensional geometries).
What why we? Since basically thanks to two articles that we published on the topic and in which we solved one of the problems opened on minimal surfaces in homogeneous spaces of more actuality in this moment. I will try to tell briefly of what it consists.
One of the basic problems on minimal surfaces is the called problem of Bernstein, which consists of classifying the minimal surfaces that are grafos especially a plane. In the space euclídeo this problem was solved in 1915 by the proper Bernstein, which demonstrated that the only ones grafos minimal entire are the planes.
One of the most studied homogeneous spaces is the Heisenberg space. This space is (topológicamente) like the usual space, but with a different metrics. Namely the way of measuring distances (and therefore everything related to the curvature) is different. In this space the vertical direction is special, and it has properties different from the horizontal directions. In the Heisenberg space he has therefore sense there appears the problem of Bernstein on that we comment previously, that is to say, to classify the minimal entire grafos under the Heisenberg space.
To solve this problem has been our biggest contribution to this theory. In 2007, and thanks to the first work of 2005, Pablo and I classify the family of grafos minimal points of the space of Heisenberg that, contrary to what it happens in the space euclídeo, is a very big family, parametrizada in terms of quadratic differential holomorfas, which are obtained from a harmonic application on the surface, but this is already another topic …
Related linkage:
This article is my contribution to the second edition of the Carnival of mathematics organized by Juan Pablo.
IntroductionThe world of the curves is a really interesting world. We can be forms of many types, from more acquaintances comoun I segment (yes, although to much they surprise a segment it is a curve in the mathematical sense of the concept) or a circumference portion, up to someone the hipopede of Eudoxo or the cuadratriz.
I feel this so extensive world of the curves we can find many with characteristics very interesting. The cicloide is, undoubtedly, one of them. It has a few very curious properties that, on having been seen, collide with our own intuition. This curve is going to be the protagonist of this article.
What is the cicloide?
Let's begin this point presenting to our friend the cicloide:
The cicloide is the curve planned by a point of a circumference (called circumference generatriz) when this one turns on a line (called straight guideline) without sliding for her.
Namely the cicloide is the curve that appears in red in the following graph:
Considering a circumference of radio and a point of the same one placed in the origin of coordinates, the equations paramétricas of an arch of the cicloide generated by this point, on having turned the circumference on the axis, are:
, with
The cicloide has been a curve much studied along the history. Already at the end of the XVIth century, Galilean he had studied this curve, obtaining certain approaches on calculations related to her (in particular on the area shut up by an arch of cicloide). Mersenne, possibly after knowing these studies of Galilean, attracted attention on the mathematicians of this epoch (we are already in the XVIIth century) towards this curve. And many were those who came to the call. So much it was the sense of expectancy created by this curve that finished with being known as the Hellene one of the geometers for the disputes quantity between mathematicians who provoked the studies related to her.
The case is that one of the first ones that they obtained turned out on the cicloide was Roberval. Mersenne proposed in 1628 the study of this curve to him and a few years later, on 1634, Roberval demonstrated that the area shut up by an arch of cicloide is exactly three times the area of the circumference that generates it. Further on also it found a method to plan the tangent one to the cicloide an any point of the same one (problem solved also by Fermat and Discardings) and realized calculation related to volumes of revolution associated with the cicloide.
Roberval did not publish these results in his moment, since he wanted to keep them in certain secret to use them as problems to be proposed the candidates for his chair. For it, when Torricelli (mathematician who also was interested by this curve) published his solutions to several of the questions solved by Roberval without mentioning him, he believed that it was a question of plagiarism. But the Torricelli studies had developed from independent form to those of Roberval. In the end the history was just with the two: Roberval was the first one in finding the solutions and Torricelli the first one in publishing them.
But the thing did not stay there. In 1658 Christopher Wren calculated that the length of an arch of cicloide is four times the diameter of the circumference that generates the above mentioned curve. And many other they were the mathematicians who dedicated part of his time to her, between that there are illustrious Pascal, Huygens, Leibniz, Newton, Jakob and Johann Bernoulli …
What properties does it have?The big interest caused by this curve comes from the curious characteristics that it possesses. Apart from the already mentioned calculations, the cicloide has two really interesting properties and that, like handyman at the beginning of the article, in certain way they commit an outrage against our intuition. Specifically there are his condition of braquistócrona and his condition of tautócrona. We are going to try to tell what they mean these two properties.
Braquistocronía
The term braquistócrona means the minor time. The problem of the braquistócrona can be enunciated of the following form:
Given a point in a plane and another point of the same placed plane vertically further down that (without going so far as to be vertically just under), to find the curve that it joins and that makes minimal the time that takes a mobile point in coming of to to be submitted to the action of the gravity
The situation of the points is something like that:
At first it would not be strange to think that this curve is a straight line (a segment in this case), since in a plane a straight line represents the shortest distance between two points. But we are not speaking about distances, but of times. Will the answer keep on being also the straight line? Let's see this video in which two appear cicloides and a segment and let's answer later:
Since it is possible to see the balls (the mobile point) they come earlier to the destination when they go down for the cicloide. Namely that in the cicloide the trip time is minor than in a segment. In fact the cicloide minimizes this trip time, that is to say, the cicloide is the braquistócrona. Onlooker: truth?
Tautocronía
The braquistocronía is not the only curious property of the cicloide. In fact it has one that is more surprising if it fits. We might enunciate it of the following way:
Let's suppose that we have a cicloide that "hangs" down and that we drop along her two balls from different points. The question is that it does not matter since we them drop points since the balls come simultaneously to the rock bottom.
This property is named tautocronía (that means the same time). We are going to see it in a video:
To finish I leave this linkage to you. It has seemed interesting to me because puncturing in each of the grids that appear we create cicloides and can see graphically two properties commented previously.
Sources:
Ah, I recognize the lion as his claw.
(Referring to Newton).
Johann Bernoulli
INFINITUM. Mathematical appointments
As we comment another day in the post on the cicloide, this curve has given place to many histories and disputes between mathematician. The phrase of this post was the end of one of them.
In 1696 Johann Bernoulli raised to the members of Royal Society two problems (at last related to the cicloide). He them was considering to be so complicated that it gave a term of six months for the presentation of the solutions and offered as I reward a valuable book of his personal collection to the one who was solving two problems. After these six months only Leibniz had solved the first one of them. In view of the results Bernoulli it gave another six months of term … but everything was still equal: neither no new solution of the first one nor no solution for the second one.
Immediately after this Leibniz suggested him to make to come these problems to Newton. This one had already spent his best moment and for it, as it seems, Bernoulli saw in this mailing a way of ridiculing him (Bernoulli was partial to Leibniz in the dispute on the invention of the calculation).
The case is that the problems came to Newton's hands one evening … and in the dawn of the same day it had already solved them. On the following morning he sent to Royal Society his solutions, but without be identifying. Bernoulli only needed to throw a glimpse to them to recognize the lion as author of the same ones.
A few Pi decimal ones
Since you will know many of you today, on March 14, it is the day of Pi. if someone does not know why, the reason is that in the Anglo-Saxon world the dates he writes himself of the form Month / day / year. Thus today would be 3/14.Every year I write something related to Pi this day. And this year is not going to be less. We are going to celebrate the day of Pi of infinite form.
Of infinite form?We are going to celebrate this day of Pi of infinite form showing diverse sums and infinite products where this wonderful number appears. We go with them:
And for exponent 6 this one:
In her the numerators of the fractions are the prime numbers except 3 and the denominators take a sum when the prime number is of the form and a subtraction when it performs the form.
Here the odd numbers appear like denominators and the signs are alternated + and - between the fractions.
And in this expression they appear in the denominators of the squares of all the odd numbers that are not multiple of 3.
I recommend the linkage to MathWorld that appears at the end of the article to see other expressions of this style which discoverer was Ramanujan.
It is the following one:
I have left to myself many expressions which protagonist is Pi. If you know someone that should not appear in this article and believe that it is important or interesting do not hesitate to write it in the comments.
Other days of Pi in Gaussianos:
Sources:
