PETERSEN GRAPH AND ICOSAHEDRON

IGOR V. DOLGACHEV

Abstract. We discuss some of the various beautiful relations between two omnipresent objects in mathematics: the Petersen graph and the Icosahedron. Both have high sym- metry and we show how these symmetry groups arise as symmetry groups of richer geometric objects: algebraic surfaces.

Petersen graph. The Petersen graph is a for his work in the invariant theory and the undirected regular 3-valent graph with 10 four-color theorem. vertices and 10 edges Kempe According to a realized the citation from Don- Petersen ald Knuth borrowed graph as from Wikipedia the the graph Petersen graph is “a whose ver- remarkable configu- tices repre- ration that serves as sent lines in a counterexample to Figure a Desargues 1. Petersen many optimistic pre- configura- Figure 3. Sir graph dictions about what tion of 10 Alfred Kempe might be true for graphs in general.” The lines and 10 author first encountered this graph on the points in projective plane with two vertices cover of the Russian translation of Frank connected by an edge if two lines do not Harary’s book [3]. meet at one of the ten points of the config- The graph uration. is named af- Recall ter a Danish from pro- mathemati- jective ge- cian Julius Pe- ometry tersen (1839- that the 1910), one of Desatgues the founders of configu- the graph the- ration is ory. In partic- Figure based on ular, he is fa- the Desar- Figure 4. Desargues mous with his gues’s The- 2. Julius configuration fundamental orem: the Petersen work on reg- sides of two perspective triangles in a projec- ular graphs. tive plane intersect at three collinear points. As often happens, it had appeared ear- A French mathematician Girard Desargues lier in the work of an English mathematician (1591-1661) is considered to be one of the Sir Alfred Kempe (1849-1922) well-known founders of projective geometry. 1 2 IGOR V. DOLGACHEV The De- differential equations. The result of the sargues blowing-up is a projective algebraic sur- graph is face: a two-dimensional projective algebraic a graph variety. Over an open subset (in Zariski with 20 ver- topology where closed subsets are the sets tices cor- of common zeros of polynomials) isomor- responding phic to an affine plane the blown-up surface to lines and looks as in the following picture: points in After we perform four blowings-up at the the Desar- Figure points p1, p2, p3, p4, we will arrive at an al- gues con- 5. Girard gebraic surface D5 which contains 10 sub- figuration Desargues sets each bijective to the projective line with with edges connecting two points if they the intersection graph isomorphic to the Pe- are realized as a point lying on a line. The tersen graph. In fact, by choosing an appro- graph admits an involutive fixed-point-free priate projective embedding of the surface 9 automorphism such that the orbit graph is in 9-dimensional projective space P one can isomorphic to the Petersen graph. realize them as lines in the projective space. The equations of the surface are very nice: 9 Quintic del Pezzo surface. Another in- we realize P as the projective space asso- carnation of the Petersen graph is as fol- ciated with skew-symmetric 5 × 5-matrices with entries in a chosen ground field (e.g. of lows. Take four points p1, p2, p3, p4 in pro- jective plane and join them by pairs to ob- real numbers R, or of complex numbers C, or any infinite field you fancy). Then take tain six lines `ij. Now each line `j intersects five general linearly independent equations three lines `ik, `j,l and `kl, where {i, j, k, l} = {1, 2, 3, 4}. We are almost there, but need and add five quadratic equations defined by four more lines with the same incidence pfaffians of five principal 4×4-submatrices of property. our skew-symmetric matrix. The resulting To create surface is a quintic del Pezzo surface, named them, for each after an Italian mathematician Pasquale del Pezzo, Duke of Caianello (1859-1936). We point pi, con- sider all di- refer to the rich theory of del Pezzo surfaces rections at to [1]. this point, i.e. Let us la- all slopes of bel 10 lines lines passing on a quin- Figure through this tic del Pezzo 6. Blowing- point. If we surface by 2- up take any line ` element sub- sets of [5] = in the plane that does not contain pi, the directions correspond to intersection points {1, 2, 3, 4, 5} as follows. of lines through pi with `. So, we obtain that all directions are naturally parameter- First any line `ij acquires ized by a projective line Ei. The process Figure that replaces a point with the set of direc- the label 7. Pascuale tions at this point is called the blowing-up {ab}, where del Pezzo of the point. It is one of the surgical tools {i, j, a, b, 5} = in algebraic geometry, symplectic geome- [5] and any ‘blown-up line’ Ei acquires the try, differential topology and the theory of label {i5}. In this labelling two vertices are PETERSEN GRAPH AND ICOSAHEDRON 3 2 joined by an edge if and only if their la- there is no a point in P with coordinates bels are disjoint subsets. Now we see that (0 : 0 : 0). This is an example of a Cre- the symmetric group S5 acts naturally by mona transformation, the foundations of automorphisms of the graph via its natural the theory of such transformations was laid action on the set [5]. Obviously, the action by an Italian mathematician Luigi Cremona is transitive on vertices and edges. (1830-1903). We are familiar with geomet- Using this one ric transformations which are now defined can easily show everywhere since our first high-school course that S5 is the in geometry. It is the inversion transforma- full automor- tion. phism group of If identify the plane with C, then the the graph. A nat- transformation is given by the formula z 7→ ural question now r/z¯. is whether we can If we re- Figure realize this sym- strict the 8. Labelled metry group as transfor- Petersen a group of auto- mation graph morphisms of an τ to the algebraic variety, open sub- in our case, the quintic del Pezzo surface. set of the The answer is yes, and it can be done in Figure line at in- the following way. First, we choose projec- 10. Inversion finity z = 2 transformation tive coordinates (x : y : z) in P in such 0, then it a way that the points p1, p2, p3, p4 have co- will be given by inversion of the coordinates ordinates (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : (x, y) 7→ (1/x, 1/y). Although our Cremona 1), (1 : 1 : 1). Then the symmetrical group transformation is not defined at the points S4 acts naturally by projective transfor- p1, p2, p3, it acts naturally on directions at mations that leave the set {p1, p2, p3, p4} each point p1, p2, p3 and hence can be lifted invariant. Namely, we let its subgroup to an action on the blow-up surface D5. The S3 act by permutations of coordinates and whole group S5 acts now on D5, and one let one more generating elelent, the trans- can show that there are no other automor- position (34) act by the transformation phisms of the del Pezzo surface D5. (x : y : z) 7→ (x − z : y − z : −z). Now it suffices to define the action of the transposi- Moduli interpretation. The group of au- tion (45) which together with S4 generates tomorphism of a quintic del Pezzo surface the whole group S5. has an obvious manifestation if we identify To do this the complement of 10 lines on D5 with the we use a trans- orbit space of ordered 5-element subsets of formation τ : 1 points in the projective line P with respect (x : y : z) 7→ to the group of projective automorphisms. (yz : xz : yz). To do so, given a point p outside the lines `ij, Although the we pass a unique conic through the points formula does p1, p1, p3, p4, p5. Using a rational parameter- make sense 1 ization of the conic by P , we find an ordered because it is set of 5 points. The orbit of this set is our Figure not defined point in the orbit space. Adding the 10 lines, at the points 9. Luigi we obtain an isomorphism between D5 and p1, p2, p3 since Cremona a compactification M0,5 of the orbit space 4 IGOR V. DOLGACHEV that can be viewed as the moduli space of in the 19the century, a general cubic surface 5-pointed stable rational curves. The con- can be given in a unique way by equations as P 3 struction can be extended to any number of above, where F has the form F = aixi . points, using a higher-dimensional general- ization of the quintic del Pezzo surface D5. If replace x5 It is constructed as the blow-up of n + 3 with −(x1 + ... + n points in P followed by further blowings- x4) and com- up of various linear subspaces spanned by pute the Hes- subsets of points of cardinality less than n. sian He(F 0) of This blowing-up realizes the moduli space the correspond- M0,n of n-pointed stable rational curves. ing polynomial It is a subject of intensive study in mod- F 0 of degree 4 ern research in algebraic geometry, and via in x1, . . . , x4, i.e. the Gromov-Witten theory in physics. Note the determinant that another compactification of the orbit of the matrix 1 Figure space of n-ordered points in P uses the geo- of second par- metric invariant theory. The work of Kempe 11. James tial derivatives was an important contribution to the invari- Sylvester of F 0, we obtain 1 ant theory of ordered point sets in P (see an equation of a quartic surface given [4]). by equations H = L = 0, where H = x ··· x (P5 1 ). 1 5 i=1 aixi Sylvester pentahedron. There is another We realization of the Petersen graph where the call 3 this vertices correspond to lines in P lying on a quartic surface. To do this we first con- sur- 3 4 face the sider P as a subspace of P with projective Hes- coordinates (x1 : x2 : x3 : x4) given by a sian of linear equation L = x1 + ··· + x5 = 0. Then F and we define the lines `ij by additional equa- denote tions xi = xj = 0 and define the points Pij it by by additional equations xk = xl = xm = 0, H(S). where i, j, k, l, m are distinct. One imme- Figure Now diately checks that each line `ij contains 12. Sylvester we see three points Pab, so this realizes the Petersen pentahedron that all graph. The group S5 acts on this set of lines by permuting the coordinates, and in lines `ij lie on H(S) and the points Pij are this way gives a geometric realization of the its singular points. The union of five hyper- symmetry of the graph. We can do more planes xi = L = 0 is called the Sylvester by realizing the lines as lines lying on a cer- pentahedron of the cubic surface S. tain surface given by equations H = L = 0, The reader will immediately see that the where H is a homogeneous polynomial of de- projection of the configuration to a plane gree 4. To do so, we consider a cubic surface from a point outside of the pentahedron be- S given by equation F = L = 0, where F is comes a Desargues configuration with per- a homogeneous polynomial of degree 3. spective triangles whose vertices are the According to a theorem of a British projections of the points P13,P14,P15 and mathematician James Sylvester (1814- P23,P24,P25. 1897), one of the principal contributor to the development of the theory of invariants PETERSEN GRAPH AND ICOSAHEDRON 5 The Hessian of fixed points is an example of an Enriques surface H(S) is surface named after an Italian mathemati- a special case of cian Federigo Enriques (1871-1946). a quartic surface The blown-up surface admits an embed- 5 whose equation is ding in P as a surface of degree 10. In this given by the de- embedding the orbits {Eij,Lij} are repre- terminant of a sented by 10 lines whose intersection graph symmetric ma- is the Petersen graph. trix whose en- Figure tries are linear Clebsch diagonal cubic sur- 13. Hessian forms in pro- face. Let us consider a special cu- quartic jective coordi- P 3 bic surface where F = xi . nates. They are named Cayley quar- It is called Cleb- tic symmetroids after a British math- sch diagonal cu- ematician Arthur Cayley (1821-1895). bic surface and it is named after Enriques sur- a German math- face. Although ematician Alfred the Hessian sur- Clebsch (1833- face H(S) is 1872). singular at the It has an ob- points Pij, we can Figure vious symmetry transform it to a 16. Alfred group isomorphic nonsingular sur- Clebsch to S5 and it acts ˜ face S by blow- Figure on its Hessian ing up the singu- 14. Arthur surface by projec- lar points, i.e. by Cayley tive automorphisms. It is the only cubic adding all tangent directions at these points surface with S5 symmetry. This realizes of germs of nonsingular curves lying on the the symmetry of the Petersen graph by pro- surface. The resulting surface is an example jective automorphisms of a cubic surface. of a K3 surface (see for their theory [5]). The Enriques surface obtained from the The Cre- Hessian of the Clebsch diagonal surface is mona involu- one of very rare Enriques surfaces whose tion defined group of automorphisms is finite [8]. In our by (x1 : ... : case the group is isomorphic to S5. So, we x ) 7→ ( 1 : obtain yet another incarnation of the Pe- 5 aix1 ... : 1 ) lifts tersen graph and its group of symmetry as a5x5 to an auto- the intersection graph of 10 lines lying on morphism of an Enriques surface with the full automor- S˜ of order 2. phism group S5. Figure It exchanges It was discovered in 1849 by Arthur 15. Federigo the directional Cayley and an Irish mathematician George Salmon (1819-1904). that a nonsingular cu- Enriques curves Eij bic surface contains exactly 27 lines. Each blown-up from points Pij with the line face x = 0 of the Sylvester pentahedron in- Lij. The intersection graph of the 20 curves i tersected by the other four faces along three Eij,Lij obtained in this way is the Desar- gues graph. The quotient of the K3-surface diagonals of, a complete quadrilateral with S˜ by the Cremona involution that acts free vertices Pij lying in the plane. 6 IGOR V. DOLGACHEV These diag- Icosahedron. onals define 15 Finally we ar- lines on the sur- rive at the icosa- face and this ex- hedron. As one plains its name. of Platonic solids The remaining it is omnipresent 12 lines form a in mathematics double-six: the since antiquity. It union of two sets Figure is a regular con- 20. Icosahedron of six skew lines, Figure vex polyhedron each intersecting 17. George with 12 vertices, 30 edges and 20 faces. 5 lines from the Salmon In the intro- other set. duction of his It is known Lectures on the that a non- Icosahedron [7], singular cu- a German math- bic surface can ematician Fe- be obtained lix Klein (1849- as a projec- 1925) writes: “A tively embed- special difficulty, Figure ded blowing- which presented Figure 18. Complete up of 6 points itself in the exe- 21. Felix quadrilat- in a projective cution of my plan, Klein eral plane no three lay in the great variety of mathematical of which are collinear and not all of them lie methods entering in the theory of the Icosa- on a conic. One realizes the 27 lines as 15 hedron.” Since Klein’s time the variety of lines joining a pair of points, 6 lines com- different methods and connections to dif- ing from directions at the six points, and ferent fields of mathematics has greatly in- 6 conics passing through all points except creased. 3 one. We can circumscribe a sphere in R The Clebsch around an icosahedron in such a way diagonal cubic that the planes passing through oppo- surfaces is the site edges cut the sphere in large cir- only cubic surface cles. This gives a regular tiling of where all 27 lines the sphere in 20 spherical triangles. are defined over Now we pass to reals and then the real projec- can be seen on tive plane by a picture of the identifying the surface. The six Figure antipodal points points must be 19. Clebsch on the sphere. chosen in a spe- surface The images of the cial way and how to choose them is related vertices become to the icosahedron. a 6-point subset Figure of 2( ). If we 22. Spherical P R blow-them up, we triangles get an algebraic 3 surface which can be embedded in P with PETERSEN GRAPH AND ICOSAHEDRON 7 the image equal to the Clebsch diagonal 20 faces of an icosahedron are mapped to cubic surface. The symmetry group of an an A5-orbit of 10 points in the plane. Each icosahedron is isomorphic to the alternating point lies on three of the 15 lines obtained group A5 ⊂ S5. from the edges. In the dual projective we The images of large circles correspond- obtain 10 lines and 15 intersection points ing to the edges are 15 lines in the plane. which realize the Petersen graph. In fact, They connect 15 pairs of the six points. one does not need to go to the dual plane. They have a peculiar property that the lines We can realize this configuration in the joining three disjoint pairs intersect at one plane itself. To do this, we use the fact that point. It is an Eckardt point on a cubic the action of the icosahedron group A5 in surface, named after a German mathemati- the projective plane is defined by its real cian F.E. Eckardt (unfortunately, no bio has 3-dimensional irreducible linear representa- been found). The number ten of Eckardt tion, and this implies that there is an in- point on a cubic surface is almost a record, variant conic in the plane. It defines the the only surface that beats it is the Fermat self-duality of the plane. The configuration cubic surface whose equation can be given of 10 lines can be obtained as the projection by the sum of four powers of coordinates in to the plane of the ten lines lying the del 3 5 P . It has 18 Eckardt points. Pezzo surface in P under an A5-equivariant 5 2 The images of 10 antipodal pairs on the projection P → P [2]. sphere corresponding to the centers of the

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