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Illustrating the Theory of NumbersMartin H. Weissman
Dept of Mathematics, University of California, Santa Cruz; [email protected]
AbstractWe describe challenges and opportunities that arise if one attempts to visualize number theory. The goal is to creategraphics which convey a theorem or argument, which tell a story, which are clear and honest, and which enableexploration. In the context of statistics, this is the mainstay of data visualization. In the number-theoretic context,problems of visualization are less familiar but tractable. From our pursuit of graphical excellence, we derive a fewbroad principles for the visualization of pure and experimental mathematics.
Introduction
Number theory is a strange old field of mathematics. It is not unified by methodology – there is algebraicnumber theory and analytic number theory and arithmetic geometry. Rather, number theory is unified byits devotion to a set of problems. At its core, these are problems about whole numbers, addition, andmultiplication. As its theoretical methods are diverse, so too are the methods of visualization of numbertheory. One must visualize large data sets, abstract algebraic concepts, and many kinds of geometry. Theauthor encountered these challenges while writing An Illustrated Theory of Numbers [10].
The study of multiplication of whole numbers quickly leads to the set of prime numbers, which I viewas the most interesting data set of mathematics. As data sets go, it belongs to the simplest class: a one-dimensional distribution of whole numbers. But this in itself raises challenges for visualization. Typicalproblems of data visualization relate to extracting a visual story from high-dimensional data. For primes,the interesting story lies in subtle properties of a one-dimensional distribution. How can one visually tell thestory of this data set? The first section addresses this challenge.
The second section addresses the visualization of a finite cyclic group: the multiplicative group F×p ofnonzero numbers modulo a prime p. Finite cyclic groups might be the least interesting, from the standpoint ofabstract algebra. But here the generators (called primitive elements) are scattered among the representatives{1, . . . , p − 1}. Abstractly simple, this group is the foundation for cryptographic protocols which, like Diffie-Hellman key exchange [3], are based on the discrete logarithm problem. How does one illustrate the structureof F×p, in order to convey its abstract simplicity along with the erratic distribution of its primitive elements?
The third section addresses a major modern (1960s) achievement of number theory: the solution ofGauss’s class-number-one problem for negative discriminants. In the language of binary quadratic forms,a theorem of Heegner, Baker, and Stark (see Stark’s exposition at [7]) proved a conjecture of Gauss [4,Art. 303].
Theorem 1 (Baker-Heegner-Stark). The following is the complete list of negative integers ∆ for which thereis a unique proper equivalence class of binary quadratic forms of discriminant ∆.
−3,−4,−7,−8,−11,−12,−16,−19,−27,−28,−43,−67,−163.
Conway’s topograph provides a visual approach to the study of binary quadratic forms. In particular,his results allow one to reinterpret the above theorem as a statement about triangles with integer side-length.This, in turn, allows one to convey the meaning of the above theorem to an audience outside of number theory.
Bridges 2018 Conference Proceedings
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Art cannot be cleanly separated from function, craft and design. An architect or designer is given a“brief” to follow, with goals (or functions) together with constraints such as audience and space and gravity(a building must stand). Their products are described as elegant, an aesthetic judgment to be sure, if form andfunction and constraint harmonize rather than conflict. The same is true of mathematical proofs, constrainedby logic and prior knowledge and directed towards a theorem.
While I began work in visualization for pedagogical reasons – teaching number theory – the exerciseof giving visual form to mathematics led to unexpected results. Though I did not think in this way at thetime, I had given myself a brief: to create images which would convey my intuition for number theory (as aspecialist) to an audience of students. Like a mathematical proof, the images had to be honest and withoutcontradiction. They had to fit on the page, and my tools were limited to programming in Python, and LaTeXwith TikZ.
To present the mathematical form visually – this imperative can drive the creation of beautiful imageswhich convey deep mathematical ideas. Satisfying the imperative is not enough. Elegance seems to comefrom a dialogue between visual aesthetics and mathematical structure. A first attempt may yield a muddledgraphic, ugly colors, awkward proportion. Fixing the visual can raise new mathematical questions – andsometimes, ideally, the visual solution requires an unexpected perspective on mathematics. There can be amoment when the visual form seems to enjoy its constraints, and the viewer can explore mathematics just bysight. That sort of elegance is a rare and valuable artistic reward.
Visualizing the Primes
Euclid (Elements, IX.20) proved that the set of prime numbers is infinite. The list
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 . . .
of prime numbers does not end. Some of the central problems of number theory, e.g., the RiemannHypothesis,address the distribution of primes. Although their small-scale structure is quite irregular, their large-scalestructure is amazingly regular. Heuristically, a whole number n has about a 1/ log(n) “chance” of being prime(where log denotes the natural logarithm). For example, among the thousand numbers between 1 000 001and 1 001 000, one would expect about 1000/ log(1 000 000) ≈ 72 primes. (The actual number is 75.)
Aminimalist approach to visualizing one-dimensional distributions is given by Brian Hayes in [5, Figure1]. His method is to simply place a horizontal line segment for each member of a data set. This allows one toperceive the difference between even and irregular spacing, data points which “repel” one another and thosethat cluster, and trends in the density of data.
From this effective minimalist starting point, one can consider the distribution of primes at differentscales: between 1 and 100, 1 and 1000, 1 and 10000. Each column of Figure 1 displays the primes between1 and 10e, with e = 2, . . . , 9.
The experiment is one of direct display. In the leftmost column, each prime is displayed as a solidbar, 10 points thick. In the second column, each prime is displayed as a line segment, 1 point thick. In thethird column, each horizontal line segment stands for a range of 10 numbers. Each segment is shaded gray,according to the number of primes within the range of 10. For example, if a range of 10 numbers contains4 primes, it is shaded with 40% black ink. By the fourth column, each horizontal line segment stands for arange of 100 numbers. By the last column, each line segment stands for a range of one million numbers. Theamount of black ink is precisely (as far as the printer can handle) the proportion of primes within the givenrange. Note that on some computer screens, a striped appearance is possible due to some “quantization” indigital image rendering.
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In the first five columns, the eye can see the transition from small-scale irregularity to large-scalesmoothness. The second column is evidently much more “choppy” than the third or fourth; one reason isthat the gaps between primes do not grow proportionally to the primes themselves. For example, the averagegap between primes near 1000 is about 7, which translates to 7 points of line-thickness in the scale of thesecond bar. But the average gap between primes near 10000 is about 9, which translates to just 0.9 points ofline-thickness in the scale of the third bar. In the fourth bar, the average gap between primes appears only0.1 points thick, and so most gaps cannot be seen. Even the largest gaps become undetectable; the largestgap between primes up to 1 million is a gap of 114 (between 492 113 and 492 227). Such a gap is only 0.114points thick in the fifth column (where it would first appear). The absence of visible gaps in the gray barsreflects the fact that the largest gaps between primes grow much more slowly than the primes themselves.
Among primes up to... 100 1000 10 000 100 000 1 000 000 10 000 000 10 000 000 1 000 000 000The largest gap is... 8 20 36 72 114 154 220 282
In the right five columns, the human eye is able to detect the slight lightening of the gray bars from left toright, and within each bar, from bottom to top, reflecting the gradual “spreading out” of primes. One couldemphasize this by using more ink on the left and less on the right. But the perfect correspondence betweenproportion of primes and proportion of black ink yields a gradualism that honestly portrays the 1/ log(n)heuristic. This also fits with Tufte’s design strategy [8] of the “smallest effective difference.”
The red horizontal and diagonal lines in Figure 1 are influenced by the cover image of Edward Tufte’sThe Visual Display of Quantitative Information [9]. They are meant to lead the reader’s eye from left to right,to imagine (for example) the primes between 1 and 100 squeezed into the bottom tenth of the primes between1 and 1000, then squeezed into the bottom hundredth of the primes between 1 and 10000.
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100 1 000 10 000 100 000 1 000 000 10 000 000 100 000 000 1 000 000 000
Figure 1: Primes between one and one billion. From The Distribution of Primes, artwork by the author.
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Visualizing a Cyclic Group
Let p be a prime number. The abelian group F×p may be described in elementary terms as follows. It consistsof the set of numbers {1, 2, . . . , p − 1} and the following rule for multiplication: the product of two elementsa, b of the set is the remainder obtained after dividing ab by p. For example, in F×7 , the product of 3 and 5is 1. We write 3 · 5 = 1 mod 7. The statement that F×p is an abelian group is the statement that this law ofmultiplication is commutative and associative, that it has an identity element (1), and that every element ofF×p has an inverse element. For example, the inverse of 3 is 5 in F×7 , since 3 · 5 = 1 mod 7.
It is a theorem of Gauss that the group F×p is cyclic. This means that there exists a primitive element –an element of F×p which generates all of F×p by repeated multiplication. For example, 3 is a primitive elementin F×7 , because repeated multiplication by 3 (mod 7) traverses all numbers in {1, . . . , 6}:
1·3−→ 3
·3−→ 2
·3−→ 6
·3−→ 4
·3−→ 5
·3−→ 1.
Here an arrow joins a to b if a · 3 = b mod 7. Finite cyclic groups are the simplest objects of grouptheory. But here the structure is concealed because the names of the elements of the group do not reflect thestructure of the group in a simple fashion. Contrast F×p with the additive group (Fp,+) which is also cyclic;a primitive element of the additive group is 1 as seen by the repeated addition (modulo 7):
0+1−−→ 1
+1−−→ 2
+1−−→ 3
+1−−→ 4
+1−−→ 5
+1−−→ 6
+1−−→ 0.
In fact, every nonzero element of the cyclic group (Fp,+) is a primitive element (a generator of the cyclicgroup), while primitive elements of F×p are distributed haphazardly.
We can interpret this as a statement about the dynamics of repeated multiplication in the group F×p. Whathappens when we start with a number a with 1 ≤ a ≤ p − 1, and repeatedly multiply by another number b(modulo p)? If b happens to be a generator, one obtains a cycle of length p − 1. If not, one obtains a cyclewhose length is a proper divisor of p − 1. In fact, cyclic groups with p − 1 elements are characterized by thefact that they have a unique subgroup of any order dividing p − 1. The dynamics encode the structure.
To visualize the cyclic group F×p, Figure 3 displays all of the dynamics without making choices. Aclose-up is below in Figure 2.
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Figure 2: A close-up from Epicycles Modulo 37 (artwork by the author). One the left, 5 is a primitiveelement in F×37. On the right, 6 generates a subgroup of order 4 (with 9 cosets).
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Every number b between 1 and 36 generates a subgroup of F×37. The full Figure 3 displays these subgroupsand their cosets for all b. Thus the figure is made of 36 subfigures, as b ranges from 1 to 36 (readingacross then down). The subfigures display a single large circle when b is a primitive element, i.e., whenb ∈ {2, 5, 13, 15, 17, 18, 19, 20, 22, 24, 32, 35}. All in all, the figure displays 36 × 36 numbers, reflecting allpossible multiplications in the group F×37. But a typical “multiplication table” with a 36x36 rectangular grid,is replaced by “dynamics table” which makes the cyclic structure evident.
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Figure 3: The dynamics of multiplication, modulo 37. From Epicycles Modulo 37, artwork by the author.
Quadratic reciprocity is a landmark theorem in number theory, first proven (in six different ways) by Gauss.Decades later, Zolotarev [11] proved quadratic reciprocity in yet another way. To start, Zolotarev observedthat if we view multiplication-by-b as a permutation of F×p, then the Legendre symbol
(bp
)coincides with the
sign of this permutation. The signs of these permutations are easily readable from a visualization like Figure3. For example,
(637
)= (−1)9 = −1, since multiplication by 6 yields 9 cycles of length four (see Figure 3),
and every length-four cycle has sign −1. The Legendre symbol is evident in the figure.An unexpected outcome of this visualization, and a source of its aesthetic appeal, is an approximate
symmetry. If x + y = 37, then an exercise in group theory demonstrates that x and y generate subgroupsof F×p of the same order, or else one subgroup has twice the order of the other. For example, in the top-leftsubfigure and bottom-right subfigure, one finds subgroups of orders 1 and 2, respectively. When x = 2 andy = 35, we find subgroups of order 36 in both cases; both are primitive elements. When x = 3 and y = 34,we find subgroups of orders 18 and 9, respectively. This “almost-equality” (up to a possible factor of 2) givesthe whole array a “near-symmetry” with respect to 180-degree rotation. Beyond aesthetics, the image allowsthe reader to explore and notice previously unseen patterns.
Illustrating the Theory of Numbers
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Visualizing Binary Quadratic Forms via Integer Triangles
Lastly, we turn to the visualization of a theorem: the complete solution to Gauss’s class-number-one problemfor binary quadratic forms of negative discriminant. This theorem is difficult to explain to a novice, but a briefsurvey follows. In his Disquisitiones [4], Gauss makes a detailed study of binary quadratic forms (BQFs,hereafter), functions Q : Z2 → Z of the form
Q(x, y) = ax2 + bxy + cy2.
Gauss requires a, b, c to be integers; Gauss required b to be even, but today this assumption is relaxed. EachBQF has a discriminant: the number ∆(Q) = b2 − 4ac familiar to those who know the quadratic equation.The number ∆(Q) is either a multiple of 4 (if b is even) or one more than a multiple of 4 (if b is odd).
Two binary quadratic forms are called properly equivalent if they are related by a change of variablesmatrix with determinant 1. To denote this, we write Q ∼ Q′ if there exists a matrix g ∈ SL2(Z) such thatQ′(x, y) = Q((x, y) · g) for all x, y ∈ Z. Properly equivalent BQFs have the same discriminant.
Gauss proved that, for any nonzero ∆, there are finitely many proper equivalence classes of binaryquadratic forms of discriminant ∆. The number of proper equivalence classes of primitive BQFs of discrim-inant ∆ is called the class number and denoted h+(∆). Here, primitive means that GCD(a, b, c) = 1. Fornegative discriminants, h+(∆) increases gradually (though not monotonically) as |∆| increases. In particular,Gauss conjectured that there are only finitely many negative integers ∆ for which h+(∆) = 1. This was solvedby Heegner, Baker, and Stark (Theorem 1): the complete list of negative ∆ for which h+(∆) = 1 is
−3,−4,−7,−8,−11,−12,−16,−19,−27,−28,−43,−67,−163.
In his Sensual (quadratic) Form [2], John H. Conway introduces the topograph – an arrangement of thevalues of a BQF on regions bounded by a ternary-regular tree in the hyperbolic plane. When Q has negativediscriminant, Conway’s topograph contains a unique well or double-well: a location at which the values areminimal, in a sense.
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Figure 4: Conway’s topographs for Q(x, y) = x2 − xy + y2 and Q(x, y) = x2 + y2. The well on the left hasthe triple of values (1, 1, 1). The double-well on the right has values (1, 1, 2).
More precisely, a well can be located on the topograph as a triple of numbers (u, v,w) surrounding a point,such that the (weak) triangle inequalities hold:
u + v ≥ w, v + w ≥ u, w + u ≥ v.
It is a theorem, essentially of Gauss, but reformulated by Conway in terms of wells, that the equivalence classof a binary quadratic form is determined by the triple of numbers surrounding its well. Conversely, every
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triple (u, v,w) of positive integers satisfying the triangle inequalities determines a unique equivalence classof binary quadratic forms. The discriminant of these BQFs is given by
∆ = u2 + v2 + w2 − 2uv − 2vw − 2wu.
Thus, to visualize the proper equivalence classes of BQFs of negative discriminants, it is equivalentto visualize all triangles with integer side-lengths (u, v,w). We must allow “degenerate triangles” – linesegments with u + v = w, for example. In Figure 5, we organize these triangles horizontally by discriminant(the number ∆ above) and vertically by their smallest side-length.
MARTIN H. WEISSMAN
3 4 7 8 11 12 15 16 19 20 23 24 27 28 31 32 35 36 39 40 43 44 47 48 51 52 55 56 59 60 63 64 67 68 71 72 75 76 79 80 83 84 87 88 91 92 95 96 99 100 103 104 107 108 111 112 115 116 119 120 123 124 127 128 131 132 135 136 139 140 143 144 147 148 151 152 155 156 159 160 163
Figure 5: Primitive triangles with integer sides u, v,w, arranged horizontally by discriminantu2 + v2 + w2 − 2uv − 2vw − 2wu and vertically by min{u, v,w}. (Artwork by the author.)
Negative discriminants,−3,−4,−7,−8,−11,−12, . . . ,−163 aremarked by vertical “pinstripes,” and numberedat the bottom. The thinnest triangles along the bottom correspond to the “principal classes” of binary quadraticforms (those with smallest side length 1). Some triangles are isosceles, and others are paired with their mirror-image. This enables the reader to compute h+(∆) by simply counting the number of triangles along a givenvertical line. For Gauss’s class number (based on proper equivalence), the non-isosceles triangles and theirmirror images are counted separately. For example, h+(−31) = 3. Isosceles triangles correspond to Gauss’sambiguous forms, capturing the “2-torsion of the class group” in modern terminology.
The rightmost line contains a single triangle, reflecting the endpoint of the Baker-Heegner-Stark theorem:−163 is the last negative discriminant with class number one. In fact, the whole theorem becomes a statementabout triangles with integer side length – if one continued Figure 5 further to the right, one would never againfind a vertical line with only one triangle on it.
The parabolic separation of light and dark reflects a classical bound: the smallest side-length of a triangleis bounded by
√∆/3. Equivalently, the smallest nonzero value of a positive-definite BQF of discriminant ∆
is bounded by√∆/3.
Illustrating the Theory of Numbers
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Principles for Visualizing Mathematics
From creating these images, I have distilled a few principles for visualizing mathematics.First: Directly show the raw data. Try to represent your data points (e.g., all multiplications in F×37)
directly, with simple dots, lines, numbers, etc. Do not bin your data or summarize until the direct portrayalhas a chance to suggest patterns.
Second: Bring mathematics into the deep design. The image in Figure 5 began with a simple concept:draw all integer-side triangles, arranged by discriminant and smallest value. When standard data-graphicselements like axes with tick marks, grids, etc., failed, mathematics guided the way. The vertical lines ofFigure 5 serve as gridlines at the same time as they exhibit the pattern of possible discriminants. To representthe smallest-value bound min{u, v,w} ≤
√∆/3, I drew a parabolic curve at first. But the curve seemed to
clutter the image without conveying the message. Eventually, the curve became the border between dark andlight, which underlies the composition of the piece. In terms of perception, the bound became the groundand the triangles the figure. Mathematics not only provided data, but deeply influenced the design.
Third: Visualize mathematics at multiple scales. When we look at an image, our eyes follow ajumpy path [6] guided by the image’s composition. Our graphics can tell a nuanced story if they operatecompositionally on a large scale and also offer fine detail (visually and mathematically). The broad structureof Figure 3 reflects the structure of a cyclic group. At a distance, one sees only circles arranged in circlesarranged in a matrix. Close up (Figure 2), the fine detail gives the actual numbers. Words or numbers mightplay a crucial role, especially at the fine scale. Multiple scales are the foundation of Figure 1. The leftmostbar contains primes as individual numerically-labeled figures; the rightmost bar displays only a light gradient.
The field of data visualization is rapidly progressing, from Tufte’s new foundations [9] to interactionand animation (e.g., the Javascript package d3.js [1]). The time is ripe for expositors of mathematics tounderstand this field and convey their ideas with a mix of text and graphics.
References
[1] M. Bostock, V. Ogievetsky, and J. Heer. “D3: Data-Driven Documents.” IEEE Trans. Visualization &Comp. Graphics (Proc. InfoVis), 2011.
[2] J. H. Conway. The Sensual (quadratic) Form, vol. 26 of Carus Mathematical Monographs. MAA,Washington, DC, 1997. With the assistance of Francis Y. C. Fung.
[3] W. Diffie and M. E. Hellman “New directions in cryptography.” IEEE Trans. on Information Theory22(6):644–654, November 1976.
[4] C. F. Gauss. Disquisitiones Arithmeticae. Springer-Verlag, New York, 1986. Translated and with apreface by Arthur A. Clarke.
[5] B. Hayes. “Computing science: The spectrum of Riemannium.” Amer. Scientist, 91(4):296–300, 2003.[6] R. Q. Quiroga and C. Pedreira. “How do we see art: an eye-tracker study.” Frontiers in Human
Neuroscience, 5(98):1–9, September 2011.[7] H. M. Stark. “The Gauss class-number problems.” In Analytic Number Theory, volume 7 of Clay
Math. Proc., pages 247–256. Amer. Math. Soc., Providence, RI, 2007.[8] E. R. Tufte. Visual Explanations. Graphics Press, Cheshire, Connecticut, 1997.[9] E. R. Tufte. The Visual Display of Quantitative Information. Graphics Press, Cheshire, Connecticut,
2001.[10] M. H. Weissman. An Illustrated Theory of Numbers. Amer. Math. Soc., Providence, RI, 2017.[11] G. Zolotareff. “Nouvelle démonstration de la loi de réciprocité de Legendre.” Nouvelles Annales de
Mathématiques. 2e série, 11:354–362, 1872.
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