{"id":84627,"date":"2023-06-26T11:04:00","date_gmt":"2023-06-26T09:04:00","guid":{"rendered":"https:\/\/aktuelles.uni-frankfurt.de\/?p=84627"},"modified":"2025-07-03T12:17:42","modified_gmt":"2025-07-03T10:17:42","slug":"a-circle-with-an-infinite-number-of-midpoints","status":"publish","type":"post","link":"https:\/\/aktuelles.uni-frankfurt.de\/en\/english\/a-circle-with-an-infinite-number-of-midpoints\/","title":{"rendered":"A circle with an infinite number of midpoints"},"content":{"rendered":"

The fascinating world of p-adic geometry<\/h4>\n\n\n\n
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Photo: amtitus\/istockphoto<\/p>\n<\/div><\/div>\n<\/div>\n\n\n\n

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The world that mathematics professor Annette Werner is exploring seems strange to us, almost absurd: in it, different numbers are of the same magnitude, and circles have an infinite number of midpoints. She is conducting research in the field of p-adic geometry \u2013 a branch of modern algebra that has made rapid advances in recent decades.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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If we want to tell someone how far one place is from another, we can do it in several ways: for example, we can express the linear distance in kilometers, in yards or in light years. However, we can also say how long it takes to get there on foot, by car or by public transport. Depending on which method and unit of measurement we choose, we obtain different values for the distance in question. It might even be the case that we are unable to convert these values intuitively into each other because the measurement systems are fundamentally different. For example, two places might be just a few kilometers apart, but the journey by bus and train nevertheless takes an eternity \u2013 in a case like this, the distance in kilometers is small, but it would take a long time when measured in terms of public transport. However, every single measurement method \u2013 whether linear, road or walking distance or distance with public transport \u2013 works and is legitimate.<\/p>\n\n\n\n

Annette Werner has started exploring an unusual but equally plausible and consistent way of measuring distances. \u201cYou could say I\u2019m just using a different ruler,\u201d she says, yet her way of measuring changes the entire character of the world she is measuring. This might sound dramatic, but it is in fact quite normal: each method of quantifying distances also produces different concepts of \u201cnear\u201d (small distances) and \u201cfar“ (large distances), like the differences between measuring by \u201clinear distance\u201d and \u201cpublic transport\u201d. What is meant by \u201cnear\u201d and \u201cfar\u201d in each case is essential to defining the character of a space. The \u201cruler\u201d that Annette Werner uses in mathematics to measure the magnitude of numbers and that makes the world of numbers seem so unfamiliar and alien to outsiders is based on prime factorization.<\/p>\n\n\n\n

Any natural number can be clearly factorized into powers of prime numbers, for example:<\/p>\n\n\n\n

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30 = 2 \u00b7 3 \u00b7 5<\/h1>\n\n\n\n

33 = 3 \u00b7 11<\/h1>\n\n\n\n

36 = 2\u00b2 \u00b7 3\u00b2<\/h1>\n\n\n\n

75 = 3 \u00b7 5\u00b2<\/h1>\n\n\n\n

76 = 2\u00b2 \u00b7 19<\/h1>\n<\/div>\n<\/div>\n\n\n\n
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This method of prime factorization can now also be used for measuring magnitude. Instead of taking the ordinary value of the number (which is the standard \u201cruler\u201d, as it were, for measuring numbers), we look at its factorization and ignore all but a certain factor called p, such as p\u202f=\u202f3 or p\u202f=\u202f907 or any other value. The reciprocal of the power of this factor p in the prime factorization of a number is known as the p-adic value of the number. Thus, p is simply written into the denominator as often as this prime number occurs as a factor in the prime factorization. It is a simple concept. Let\u2019s calculate the p-adic value of the above numbers when p\u202f=\u202f5:<\/p>\n\n\n\n

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|30| = 1\/5<\/h1>\n\n\n\n

(because 5 occurs exactly once as a prime factor)<\/p>\n\n\n\n

|33| = 1<\/h1>\n\n\n\n

(because 5 does not occur as a prime factor, and 50 = 1)<\/p>\n\n\n\n

|36| = 1<\/h1>\n\n\n\n

(likewise)<\/p>\n\n\n\n

|75| = 1\/52 = 1\/25<\/h1>\n\n\n\n

(because 5 occurs exactly twice as a prime factor)<\/p>\n\n\n\n

|76| = 1 (see above)<\/h1>\n\n\n\n

If the prime number p = 3, the p-adic values of the above numbers are:<\/p>\n\n\n\n

|30| = 1\/3
|33| = 1\/3
|36| = 1\/32 = 1\/9
|75| = 1\/3
|76| = 1<\/h1>\n<\/div>\n<\/div>\n\n\n\n

This way of measuring the magnitude of a number seems bizarre: if, for example, p = 3, then the numbers 30, 33 and 75 are of equal magnitude, and 36 is smaller than 33. All this contradicts our understanding of numbers and magnitudes. Apart from that, however, there is nothing at all strange about the p-adic value. On the contrary, as Annette Werner explains: \u201cWe can show that the p-adic value works in principle in the same way as the notion of distance, which we use in our everyday lives.\u201d This means that it is a technically non-contradictory method that also fulfils certain conditions which are useful for measuring length. Essentially, therefore, the p-adic value functions in the same way as other methods for measuring length. \u201cMost people are content with the ordinary ruler,\u201d says Werner with a smile, \u201cbut we mathematicians are not. This shows very nicely how we think. We ask ourselves what is generally behind the principle of \u2018distance\u2019 and \u2018ruler\u2019 and what the mechanisms are that all rulers and distances have in common.\u201d<\/p>\n\n\n\n

The concept of the p-adic value, that is, expressing the magnitude of a number by one of its prime factors p, not only works for natural numbers but can also be applied to fractions. Incidentally, what you choose as p is irrelevant because every choice leads to a world that works according to the same laws. \u201cIt makes no sense at all, for example, to look only at p\u202f=\u202f17,\u201d explains Annette Werner, \u201call p are the same for us.\u201d That is why p-adic numbers and values are referred to using the abstract p instead of a specific number: the structure of the numbers and values remains the same, regardless of whether p\u202f=\u202f11 or p\u202f=\u202f599 or another prime number is chosen. Mathematicians consider all cases at the same time rather than specific situations by using p as a flexible placeholder.<\/p>\n\n\n\n

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There is even more to the p-adic world than an equivalent to our familiar fractions: just as the gaps between our classical fractions are filled by irrational numbers such as \u03c0, e or \u221a2, the gaps between the p-adic fractions can also be filled so that a dense line of numbers results, very similar to the real numbers \u2013 except that here they are slightly different numbers, and distances are calculated differently here.<\/p>\n\n\n\n

From these new numbers and distances, we now arrive at Annette Werner\u2019s specialist field, p-adic geometry, by using these new numbers as a basis for geometric concepts. In a first step, we can analyze equations, for example, and examine the figures that solve the equations, exactly as we do with conventional real numbers. For example, the real equation x\u202f=\u202fy is solved by all points (x, y) in the two-dimensional coordinate system in which x and y are identical; these points form a straight line. The equation x2<\/sup>\u202f+\u202fy2<\/sup>\u202f=\u202f1, on the other hand, produces a circle. Examining which equations describe which figures and which abstract properties of the equations are reflected in which properties of the corresponding figures is one of the classic challenges in geometry.<\/p>\n\n\n\n

By contrast, p-adic numbers produce a different kind of geometry than the one we are accustomed to from experience because they behave differently, and this leads to curious results: in p-adic geometry, for example, each circle has an infinite number of midpoints. \u201cYou can\u2019t imagine that,\u201d says Annette Werner reassuringly, \u201cbut you don\u2019t have to, you can examine it with mathematical methods.\u201d We cannot begin to imagine an infinite number of midpoints, and it sounds absurd, but at the end of the day it is the clear and incontestable result of a simple calculation: the midpoint of a circle \u2013 in our geometry as well as in other geometries \u2013 is defined as the point that has the same distance from all points on the circumference, and with the p-adic way of measuring distances, all points inside the circle have the same distance from all points on the circumference. Thus, beyond all imagination, all points in the circle, no matter where they lie, are quite simply the midpoint.<\/p>\n\n\n\n

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You can arrive at p-adic numbers through whole numbers that have the same remainder classes. If you divide natural numbers by 3, for example, the remainder is either 0, 1 or 2:<\/p>\n\n\n\n

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0 modulo 3 = 0<\/h4>\n\n\n\n

1 modulo 3 = 1<\/h4>\n\n\n\n

2 modulo 3 = 2<\/h4>\n\n\n\n

3 modulo 3 = 0<\/h4>\n\n\n\n

4 modulo 3 = 1<\/h4>\n<\/div>\n<\/div>\n\n\n\n

How to calculate \u201cmodulo 3\u201d is explained, for example, at https:\/\/tinyurl.com\/ModuloOperations All numbers that produce the same remainder after dividing by 3 (yellow circles in the top grey circle) are referred to as the remainder class.<\/p>\n\n\n\n

The grey circle in the middle contains the remainder if you divide natural numbers by the second power of 3, i.e. 32<\/sup> = 9: 0, 1, 2, 3, 4, 5, 6, 7 and 8 in the orange circles.<\/p>\n\n\n\n

Three orange circles are in a yellow circle if the corresponding numbers have the same remainder class modulo 3, for example:<\/p>\n\n\n\n

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2 modulo 3 = 2<\/h4>\n\n\n\n

5 modulo 3 = 2<\/h4>\n\n\n\n

8 modulo 3 = 2<\/h4>\n<\/div>\n\n\n\n

Therefore, in the top yellow circle (where previously only the 2 was) there are now smaller, orange circles with 2, 5 and 8, all of which have the remainder 2 modulo 3 \u2013 although they are different remainders of modulo 9.<\/p>\n\n\n\n

The game continues with the third grey circle and the third prime power 33<\/sup>: if we look for remainders modulo 33<\/sup> = 27, then each orange circle is divided again into three circles. 8, 17 and 26 are entered in the circle at the very top containing 8. These three numbers all have the remainder 8 modulo 9, but different remainders modulo 27. That is why they are in the same orange circle, but in different blue ones.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Another simple calculation, which students can also do as homework, shows that in the p-adic world each triangle has at least two sides of the same length, or in other words: triangles with three sides of different lengths cannot exist. \u201cThis all sounds strange,\u201d admits Annette-Werner, \u201cbut that\u2019s only because the words \u2018triangle\u2019 and \u2018circle\u2019 evoke images in our heads that are based on our experience, and these are specific cases that do not reflect the p-adic situation.\u201d<\/p>\n\n\n\n

Triangles with at least two equally long sides, circles with an infinite number of midpoints, large numbers with small values \u2013 from our everyday experience, all this might seem absurd to us, but the p-adic numbers form an orderly, plausible, regular world, within which everything is organized according to coherent and logical laws. What makes this world so peculiar for us is only that it is populated by numbers that we are not familiar with in our everyday lives, and that the rules which they are based on are also different from those we are accustomed to from the numbers we use every day.<\/p>\n\n\n\n

Geometry is more than just investigating a formal equation and its set of solutions. For example, several equations can be analyzed together. We can also put a variable in a sum with an infinite number of summands and ask ourselves when and in what context this sum, which is nothing more than a formally expressed object, acquires a purposeful meaning, what properties it has, what it has in common with the infinite sums we know from the world of real numbers, and much more.<\/p>\n\n\n\n

\u201cI like to peel away the shell and get to the kernel,\u201d says Annette Werner, \u201cfor example, I take a concept and see whether it works in a different context in the same way, or somehow differently. Why? Because that\u2019s how a mathematician\u2019s brain works!\u201d She explains: \u201cWe ask ourselves what language we can use to understand an object and what it means in other languages.\u201d Annette Werner has recently succeeded in transferring parts of what is known as the Simpson correspondence to the p-adic world. This work is concerned with mathematical structures called Higgs bundles and how they interact with the topology of manifolds.<\/p>\n\n\n\n

A manifold is an abstract generalization of a surface, for instance a pretzel or a bagel. Topology is the theory of spatial objects and their tear-free deformations. Imagine, for example, that the pretzel is made of rubber and you pull on the loops without tearing them. The pretzel then changes shape, but the number of holes remains the same. Conserved quantities, such as the number of holes, which do not depend on the specific shape of an object but reveal something about its basic structure, are of particular interest in mathematical research. After all, Higgs bundles are related to the Higgs boson, the famous elementary particle, and two scientists were jointly awarded the Nobel Prize in Physics in 2013 for its theoretical investigation (after experimental physics research at the CERN particle accelerator in Geneva in 2012 had succeeded in providing experimental evidence that this particle actually exists).<\/p>\n\n\n\n

Annette Werner has transferred this concept \u2013 the interaction of Higgs bundles with manifolds \u2013 to p-adic geometry. \u201cI\u2019m investigating the p-adic \u2018cousins\u2019 of these Higgs bundles to find out what they can tell us about the geometry and topology they encounter there.\u201d In her work, Werner has used the theory of perfectoid spaces developed by the mathematician Peter Scholze at the University of Bonn, among other theories. In 2018, Scholze was awarded the Fields Medal for his work, one of the most prestigious prizes in the field and known also as the \u201cNobel Prize of Mathematics\u201d. \u201cGetting used to these methods is hard work when you are no longer young,\u201d recalls Annette Werner, \u201cbut I can use these methods to illuminate and prove things that were previously a closed book to me.\u201d<\/p>\n\n\n\n

Despite the close relationship to the famous Higgs boson, Annette Werner believes that her research is unlikely to interest anyone at CERN. After all, it\u2019s fundamental research. \u201cIt doesn\u2019t bother me whether what I\u2019ve discovered can be applied to a specific problem, that\u2019s not what motivates me. But I know that I\u2019m laying a solid foundation for our science with my work, and we\u2019ll need this sooner or later to solve concrete problems in the future.\u201d<\/p>\n\n\n\n

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IN A NUTSHELL<\/strong><\/p>\n\n\n\n