1 Introduction

This paper began with a problem, dear to the heart of Paola Biondi, for which she found in 2006 a direct proof.

Proposition 1.1

(Biondi [74]) An oval O in \(\textrm{PG}(2,q)\), q even, contains three points, such that, for every choice of fourth point on the oval, the diagonal line of the four points is tangent to the oval if and only if O is a conic.

When Paola Biondi retired, she showed the result to Pia Lo Re, saying that she was also interested in related problems. In turn, when Pia Lo Re retired, she suggested looking at this area to Nicola Durante. Then Nicola Durante asked the question to Tim Penttila. It turned out that he had done in 1997 some related work [89, Theorem 4.3].

Eventually, it became clear that the question was related to the study of harmonic ovals in projective planes of even order, initiated by Cherowitzo in 1985 [17], as an extension of the ideas of Ostrom in 1955 and Ostrom in 1981 [72, 73] for harmonic ovals in projective planes of odd order.

Now we open Sects. 6, 7 below with arguments showing that, in projective planes admitting an involutory central collineation g, the local specialisation of harmonic to the centre and axis is equivalent to the oval being stabilised by g, provided that the centre is not on the oval, is on a tangent line, is not on all tangent lines and the axis is also appropriately situated relative to the oval.

Thus, when approached using the methods of symmetry, these problems find their natural setting in projective planes that either have as many involutory elations as possible or have as many involutory homologies as possible. These are the Moufang planes (Theorem 2.12).

Finite Moufang planes are Pappian (by the Artin-Zorn-Levi theorem, see, e.g. [90]), so the point of the extension of the problem from finite Pappian planes to Moufang planes lies in the infinite setting. In infinite planes there are ovals which neither have their tangent lines forming an oval in the dual plane nor have all their tangent lines concurrent. (The latter kind of oval is called central.) Moreover, even conics in Pappian planes over imperfect fields of characteristic two have two kinds of point not on the oval (and not on all tangent lines), those on a tangent line, and those on no tangent line, as first observed by Segre in 1951 [9]. Viewing the definition of a harmonic oval as a synthetic substitute for an oval admitting many involutory central collineations, and noting that Ostrom restricted the analogue of the centre of the collineation to be a point lying on a tangent line to the oval, there are two choices for the extension of the concept of harmonic central oval to infinite Fano planes: one where the analogue of the centre of the collineation is, as for Ostrom, a point lying on a tangent line to the oval, which we call harmonic and other where it is allowed to be any point not on the oval other than the nucleus of the oval, which we call strongly harmonic.

It turns out that harmonic central ovals of Moufang planes are precisely the Moufang ovals of Tits in 1962 [40] and Hartmann in 1981 [25], while amongst Moufang planes, only the Pappian planes contain strongly harmonic ovals, and these ovals are precisely the conics.

Having observed a connection between strongly harmonic ovals and ovals satisfying one of the two four-point degeneration of Pascal’s theorem, this leads us to consider the other four-point degeneration of Pascal’s theorem and the three-point degeneration of Pascal’s theorem, in the final section.

2 Projective planes and central collineations

Our main aim is to study harmonic ovals in symmetric projective planes, so we begin by surveying the projective planes in question, and their characterisations via symmetry relevant for us.

A projective plane is Pappian if it satisfies Pappus’ theorem: if A, B, C are distinct collinear points and \(A', B', C'\) are distinct collinear points (and none of them is one both lines), then \(AB' \cap A'B\), \(AC' \cap A'C\), \(BC' \cap B'C\) are collinear. This goes back to Pappus of Alexandria, and his Collection from c.340 A.D (see [45]). A projective plane is Desarguesian if it satisfies Desargues’ theorem: if two triangles are in perspective from a point, then they are in perspective from a line. This goes back to Desargues in 1640 (see [22]). A projective plane is Moufang if it satisfies the little Desargues’ theorem (which is Desargues’ theorem in the special case where the point of perspectivity is incident with the line of perspectivity). A projective plane is a Fano plane if the diagonal points of every quadrangle are collinear.

Theorem 2.1

(Hessenberg 1905) A Pappian projective plane is Desarguesian.

A case overlooked by Hessenberg in the proof was dealt with by Cronheim in 1953 [7].

Theorem 2.2

(Hilbert-Veblen-Young) A projective plane is Desarguesian if and only if it is coordinatised by a division ring.

A projective plane is Pappian if and only if it is coordinatised by a field.

Corollary 2.3

Desargues’ theorem implies its dual, which is also its converse.

An alternative ring is an abelian group R under addition, together with a product which satisfies both distributive laws, and such that \(a(ab) = (aa)b \) and \((ba)a = b(aa)\), for all \(a, b \in R\). (Note that multiplication need not be associative.) An alternative division ring is an alternative ring R with multiplicative identity \(1 \ne 0\) such that, for all \(a \in R, b \in R\), with \(a \ne 0\) the equations \(ax = b\) and \(ya = b\) have unique solutions \(x,y \in R\).

For the oft-told story of the error in Moufang’s classification and Hall’s correction of it see the first footnote on page 176 of [41]. The second footnote on the same page mentions that Pickert introduced the term Moufang plane in [33], following the advice of his supervisor, Max Dehn, and after also consulting with Reinhold Baer. For more on Ruth Moufang, see [18] and [4].

Theorem 2.4

(Moufang-Hall [36, 80, 81]) A projective plane is Moufang if and only if it is coordinatised by an alternative division ring

Corollary 2.5

A Moufang plane is either a Fano plane or has the diagonal points of every quadrangle not collinear. In the first case, involutory central collineations are elations, and in the second case, they are homologies.

Proof

The centre of the alternative division ring is a field. The first case is when the field has characteristic two and the second case is when the field doesn’t have characteristic two. Alternatively, the collineation group of a Moufang plane is transitive on quadrangles. \(\square \)

Nonassociative alternative division rings were classified (as octonion algebras that are not split) by Skornyakov in 1950, Bruck and Kleinfeld in 1951 and by Kleinfeld in 1951. For octonion algebras and for planes coordinatised by octonion algebras, see the paper by Faulkner [26, Chapter 4] and [26, Chapter 12].

Theorem 2.6

(Skornyakov-Bruck-Kleinfeld [14, 48, 84]) A projective plane is Moufang if and only if it is coordinatised by an octonion algebra that is not split.

All known Fano planes are Moufang planes. It is an open problem to classify infinite Fano planes. Finite Fano planes were classified by Gleason in 1956 [31]. For a proof see [26, Theorem 4.13]

Theorem 2.7

(Gleason [31]) A finite Fano plane is Pappian.

The intimate connection between special cases of Desargues’ theorem and central collineations was delineated by Baer in 1942.

A projective plane is \((V,\ell )\)-Desarguesian if every pair of triangles in perspective from V with two pairs of corresponding sides meeting on \(\ell \) is in perspective from \(\ell \). A projective plane is \((V,\ell )\)-transitive if for every pair P, Q of points not on \(\ell \) and collinear with V, but distinct from V, there is a collineation g with centre V and axis \(\ell \) such that \(P^g=Q\).

Theorem 2.8

(Baer [78]) A projective plane is \((V,\ell )\)-Desarguesian if and only if it is \((V,\ell )\)-transitive.

Corollary 2.9

(Baer [78]) A projective plane is Desarguesian if and only if it is \((V,\ell )\)-transitive for all points V and lines \(\ell \).

Colloquially, that says that Desarguesian planes are precisely those that have as many central collineations as possible.

Corollary 2.10

(Pickert [33]) A projective plane is Moufang if and only if it is \((V,\ell )\)-transitive for all incident points V and lines \(\ell \).

Colloquially, that says that Moufang planes are precisely those that have as many elations as possible.

A translation plane is a projective plane such that there is a line \(\ell \) (called a translation line) such that the plane is \((V,\ell )\)-transitive for all points V on \(\ell \).

Based on the work of Skornyakov in 1951 [85] and San Soucie in 1955 [86], we have:

Theorem 2.11

(Skornyakov-San Soucie [85, 86]) A projective plane is a Moufang plane if and only if it is a translation plane with respect to two different translation lines.

For a proof of the previous theorem see the book of Hughes-Piper of 1973 [39, Theorem 6.16]

Theorem 2.12

Let \(\pi \) be a projective plane. If for every flag (C, a) of \(\pi \) and every pair A, B of points distinct from C and collinear with C, there is an involutory elation with centre C and axis a interchanging A and B, then \(\pi \) is a Moufang Fano plane, and conversely. If for every antiflag (C, a) of \(\pi \) there is an involutory homology with centre C and axis a interchanging A and B, distinct from C, \(A\ne B\), A, B, C collinear with neither A nor B is on the line a, then \(\pi \) is a Moufang plane that is not a Fano plane, and conversely.

Proof

This is a straightforward application of Lemma 4.21 of Hughes and Piper in 1973 [39] that the product of two involutory homologies with the same axis is an elation. \(\square \)

3 Ovals and their tangent lines

In moving away from the finite setting, it turns out to be impossible to prove that the tangent lines to an oval are well-behaved, and so necessary to hypothesise this. Here we briefly survey this phenomenon. For the proofs of next two theorems see the book of Veblen-Young of 1910 [91, Chapter V, Theorem 11, p. 116] and the paper by Mitchell of 1910 [66, Theorem 4, p. 5] and Dickson of 1914 [23, Part I, pp. 66–67]

Theorem 3.1

(Veblen-Young [91]) The tangent lines to a conic in a Pappian projective plane that is not a Fano plane form a conic in the dual plane.

Theorem 3.2

(Mitchell-Dickson [23, 66]) The tangent lines to a conic in a finite Pappian projective plane that is a Fano plane are concurrent.

The founding papers on ovals in finite projective planes are from the 1940s. An oval of a finite projective plane is a set of points of the same cardinality as a line of the plane (considered as a set of points) with no three points collinear. A polarity of a projective plane is an incidence-preserving map of order two interchanging points and lines. A point is absolute with respect to a polarity if it is incident with its image under a polarity.

Theorem 3.3

(Baer [79]) A polarity of a finite projective plane has at least as many absolute points as the number of points on a line with equality if and only if the set of absolute points is an oval.

Corollary 3.4

A conic of a Pappian plane of odd order is an oval.

The order of a finite projective plane is one less than the cardinality of a line.

Theorem 3.5

(Bose [11]) A set of points, no three collinear, of a finite projective plane has cardinality at most two more than order, if the plane has even order and one more than the order, if the plane has odd order.

When equality occurs in the first bound, the set is called a hyperoval.

Theorem 3.6

(Bose [11]) The union of a conic of a Pappian plane of even order and the intersection of its tangent lines is a hyperoval.

Theorem 3.7

The tangent lines to an oval in a projective plane of odd order form an oval in the dual plane.

Theorem 3.8

The tangent lines to an oval in a projective plane of even order are concurrent.

Theorem 3.9

The tangent lines to a conic in a Pappian projective plane that is a Fano plane are concurrent. All lines on the point of concurrency of the tangent lines are tangent lines if and only if the plane is coordinatised by a perfect field.

Of course, the most famous result of ovals is:

Theorem 3.10

(Segre [10]) An oval of a Pappian projective plane of odd order is a conic.

Qvist’s results have finiteness as a necessary condition: we now turn to ovals in infinite projective planes.

Tits in 1962 [40] defined an oval in a (not necessarily finite) projective plane as a set O of points, no three collinear, such that for all points P in O exactly one line t has \(t \cap O=\{P\}\). This gives the same objects as the earlier definition when the plane is finite.

We can also define a hyperoval in a (not necessarily finite) projective plane as a set H of points such that every line meets H in 0 or 2 points. Again, this gives the same as the earlier definition when the plane is finite. Mazurkiewicz in 1914 [63] constructs a hyperoval in the real affine plane and so the real projective plane using transfinite induction, with a method that works for any infinite projective plane. Barlotti in 1967 [1] shows that, given two disjoint sets E and T of lines of an infinite projective plane \(\pi \) such that if a line does not belong to E, there are infinitely many points on it through which no lines of \(E \cup T \)pass, other than the line itself if it belongs to T and there is at least one line that does not belong to E, then there is an oval of \(\pi \) for which E is the set of external lines and T is the set of tangent lines. This was further generalised by Krier in 1968 [52] and in 1969 [51, Theorem 2.1]. Krier defines a semi-oval (for us, in a projective plane \(\pi \)) to be a pair \((\Omega , T)\), where \(\Omega \) is a set of points of \(\pi \), and T is a set of lines of \(\pi \) such that no three points of \(\Omega \) are collinear, and there is a bijection \(\phi \) between \(\Omega \) and T such that \(\phi (P) \cap \Omega =\{P\}\), for all \(P \in \Omega \). So T is a set of (nominated) tangent lines to the arc \(\Omega \). A very special case of his Theorem 2.1 says that every finite semi-oval of an infinite projective plane can be extended to an oval of that plane in such a way that the set of nominated tangent lines T to the semi-oval are tangent lines to the oval. Since finite arcs can have three tangent lines at distinct points concurrent, four tangent lines at distinct points concurrent, et cetera, it follows that ovals can have three tangent lines concurrent, four tangent lines concurrent, et cetera. Of course, such conclusions could also be drawn from Barlotti’s result.

See also Barlotti-Strambach [2, Theorem 2.1 and 2.2], where even up a symmetry hypothesis, such ovals are constructed.

The upshot of all these results is that there is no hope of controlling the tangent lines to an oval in an infinite projective plane in any meaningful sense. Thus we’ll need hypotheses on the tangent lines to an oval on occasions. Ostrom in 1981 [73] suggests assuming either that the oval is central or that the tangent lines to the oval form an oval in the dual plane, presumably because that’s how conics behave.

4 Symmetric ovals

Since our main aim is to show that in symmetric projective planes, various versions of a harmonic condition on ovals imply that the ovals admit involutorial central collineations, and thereby classify the ovals, in this section, we survey results classifying ovals in term of admitting involutorial central collineations,

The earliest result (for real conics, in an affine setting) is due to Wallis in 1655 [94], who proves this by laborious calculations in analytic geometry.

Theorem 4.1

Given an affine conic K and two points P, Q on K, there is an affinity fixing K and interchanging P and Q.

Stated projectively, it says that:

Theorem 4.2

Every pair of points on a conic can be interchanged by an involutory central collineation fixing the conic.

It is worth remarking that the fact that a circle is fixed by a reflection in a diameter and a half-turn about the centre implies that a real conic, being the image of a circle under a central projection, admits involutory central collineations, both with axis secant to the conic and with axis external to the conic, and that this kind of argument is characteristic of Poncelet in 1822 [42].

An oval O of a projective plane \(\pi \) is symmetric about a line a if, whenever A and B are distinct points of O not on a, there is an elation of \(\pi \) with axis a interchanging A and B and leaving O invariant. If this holds, a is called a symmetry axis of O. (Hartmann [25, Definition 1].)

It is clear that Hartmann’s definition is motivated by conics in Pappian Fano planes, for which every line on the nucleus is a symmetry axis.

Lemma 4.3

(Hartmann [25]) A symmetry axis of an oval is tangent or external to the oval. Moreover, the tangent lines to an oval with an axis of symmetry are concurrent in a point (called the nucleus of the oval) on the symmetry axis.

Proof

Every pair of tangents to O at points not on a symmetry axis a are interchanged by an elation g with axis a; thus g fixes the intersection of that pair of tangents, which is therefore on a. Hence all tangents to O at points not on a are concurrent in a point on a. The tangent to any point of O on a is fixed by any elation with axis a leaving O invariant. So a is not secant to O.

\(\square \)

An oval is a translation oval if it has a tangent line as a symmetry axis. An oval is a Moufang oval if every tangent line is a symmetry axis. (Hartmann [25, Definition 2])

Buekenhout in 1976 [15] introduced the term Moufang oval to resolve the conflict between the conflicting use of the term translation oval by Segre [82] and Tits [40].

Segre in 1957 constructed translation ovals that are not conics in [82] in finite Pappian planes of even order.

Theorem 4.4

(Segre [82]) In \(\textrm{PG}(2,2^h)\), if \((i,h)=1\) and \(1<i<h-1\), then \(\{(1,x,x^{2^i}):x \in GF(2^h)\} \cup \{(0,0,1)\}\) is a translation oval that is not a conic, nor a pointed conic.

Lemma 4.5

An oval is a Moufang oval if and only if it is symmetric with respect to two distinct tangent lines.

Proof

The group generated by the elations leaving invariant the oval with symmetry axes the two tangent lines acts transitively on the points of the oval, and so on the tangent lines to the oval. \(\square \)

Corollary 4.6

The group of a Moufang oval acts 2-transitively on the points of the oval.

Examples of Moufang ovals

  1. (1)

    A conic in a Pappian projective plane over a field of characteristic 2 is a Moufang oval [Tits [40], 2.4.1]

  2. (2)

    Let F be a field of characteristic 2 and \(K=F(t)\) be a transcendental extension of F. Let \(L=F(t^2)\). Every element of K can be written as \(x+yt\), with \(x,y \in L\). Let a, b be elements of K, linearly independent over L. Then \(\{(1,x+yt,ax^2+by^2):x,y \in L\} \cup \{(0,0,1)\}\) is a Moufang oval of \(\textrm{PG}(2,K)\), which is a conic if and only if \(b=at^2\). [Tits [40], 2.4.1]

  3. (3)

    Let \(K_0\) be a field of characteristic 2 with an automorphism \(\alpha \) of order 2. Let \(K=K_0(t;\alpha )\) be the skewfield with \(kt=tk^\alpha \) for \(k \in K_0\) (Cohn [19, p. 441]). Then \(=K_0(t^2)\) is a commutative subfield of K, and every element of K can be written as \(x+yt\), with \(x,y \in L\). Then \(\{(1,x+yt,ax^2+by^2):x,y \in L\} \cup \{(0,0,1)\}\) is a Moufang oval of \(\textrm{PG}(2,K)\). (Hartmann [25, p. 192]

  4. (4)

    Let \(K_0\) be a field of characteristic 2 and \(L=K_0(u,v,w)\), for u, v, w independent transcendentals over \(K_0\). Let K be the Cayley algebra over L ( [33, p. 177]). Let \(\{b_1,b_2,b_3,b_4,b_5,b_6,b_7,b_8\}\) be a basis for K over L with \(b_1=1\).Every element of K is of the form \(b_1x_1+b_2x_2+b_3x_3+b_4x_4+b_5x_5+b_6x_6+b_7x_7+b_8x_8\) with \(x_i \in L\). Then \(\{( b_1x_1+b_2x_2+b_3x_3+b_4x_4+b_5x_5+b_6x_6+b_7x_7+b_8x_8,x_1^2+ux_2^2+vx_3^2+wx_4^4+uvx_5^2+uwx_6^2+vwx_7^2+uvwx_8^2):x,y \in L\} \cup \{(\infty )\}\) is a Moufang oval of the Moufang plane over K. (Hartmann [25, p. 192])

  5. (5)

    Further examples of Moufang ovals in Desarguesian planes that are not Pappian appear in Yaqub in 1990 [95], Examples on pp. 266–267.

Note that a conic in a Pappian plane over an imperfect field has a symmetry axis that is external to the conic.

Do there exist ovals in Pappian planes that are not conics which have a symmetry axis that is external to the oval?

Theorem 4.7

(Hartmann [25], Satz 3.2) A Moufang oval has all lines on its nucleus tangent to the oval if and only if the plane is Pappian over a perfect field of characteristic two and the oval is a conic

Two points A, B of an oval O of a projective plane \(\pi \) are symmetric if there is an involutorial collineation of \(\pi \) stabilising O with fixed points on O being precisely A and B.

Theorem 4.8

(Mäurer [60], Satz, pp. 434–435) An oval in a Moufang plane has every point pair symmetric if and only if the plane is Pappian, not a Fano plane, and the oval is a conic.

The first local version of this result is due to Buekenhout in 1969.

Theorem 4.9

(Buekenhout [29], 4.5) An oval in a Pappian plane stabilised by a central collineation with centre any point not on the oval and on a fixed secant line is a conic.

Mäurer in 1980 extended this to Moufang planes.

Theorem 4.10

(Mäurer [59], 4.1, Theorem 1) An oval in a Moufang plane stabilised by a central collineation with centre any point not on the oval and on a fixed secant line is a conic in a Pappian plane.

Theorem 4.11

(Mäurer [60]) An oval O in a Moufang plane with a point A of O stabilised by a central collineation with axis AB for all \(B \in O\) with \(B \ne A\), is a conic in a Pappian plane that is not a Fano plane.

Theorem 4.12

(Mäurer [61]) An oval O in a Moufang plane with a point A of O stabilised by a central collineation with centre any point not on the oval and on a fixed tangent line is a conic in a Pappian plane that is not a Fano plane.

Theorem 4.13

(Mäurer [61]) An oval in a Pappian plane that is not a Fano plane with a point A of O stabilised by a central collineation with centre any point on a fixed external line is a conic.

Theorem 4.14

(Payne [76]) In a finite Fano plane, a translation oval is equivalent to \(\{(1,x,x^{2^i}):x \in GF(2^h)\} \cup \{(0,0,1)\}\), for some i with \((i,h)=1\).

Proof

By Gleason’s Theorem 2.7, the plane is Pappian. Now we have Payne’s hypotheses, and can apply the main result of [76]. \(\square \)

Theorem 4.15

(Tits-Payne [40, 76]) In a Pappian projective plane of even order, a Moufang oval is a conic.

5 Harmonic ovals in projective planes

In 1955, inspired by the harmonic property of conics first pointed out by Apollonius of Perga in his book Conics around 200 B.C. [6, Book I, Proposition 36], Ostrom initiated the study of harmonic ovals [72]. In 1981, Ostrom clarified and broadened his definition of a harmonic ovals [73].

Definition 5.1

(Ostrom [73]) An oval O in a projective plane of odd order is harmonic if, whenever U is the intersection of the tangent line to O at X and the tangent line to O at Y and secant lines \(\ell ,m\) on U meet O at A, B and \(A',B'\), respectively, \(AB' \cap A'B\) is on XY.

The origin of Ostrom’s definition is a theorem of Brianchon from in 1817 [64, Article XX], proved there for real conics:

Theorem 5.2

(Rigby [43], 4.5 (ii)) The diagonal triangle of a quadrangle inscribed in a conic in a Pappian plane that is not a Fano plane is self-polar.

Ostrom in 1955 proved that a conic of a Pappian projective plane of odd order is harmonic.

Theorem 5.3

(Ostrom [72], Theorem 3.5.1) A conic of a Pappian projective plane of odd order is a harmonic oval.

An oval O in a projective plane of even order is harmonic if, whenever the point U is not on the oval, and secant lines \(\ell ,m\) on U meet O at A, B and \(A',B'\), respectively, \(AA' \cap BB',AB' \cap BA', AA \cap BB\), and \(A'A' \cap B'B'\) are collinear (and on the same line for any pair of secants on U). (AA is the tangent to O at A, et cetera.) (Cherowitzo [17]).

We delay tracing the origin of Cherowitzo’s definition until the final section of the paper.

Cherowitzo in 1985 proved that a conic of a Pappian projective plane of even order is harmonic.

Theorem 5.4

(Cherowitzo [17], proof of Theorem 2) A conic of a Pappian projective plane of even order is a harmonic oval.

Theorem 5.5

(Cherowitzo [17], Theorem 2) Let O be a harmonic oval in a projective plane \(\pi \) of even order n. Then n is a power of 2.

Ostrom in 1981 [73, pp. 187–188] pointed out the need, when studying harmonic ovals to assume that either no three tangent lines are concurrent or that the oval is central.

Ostrom’s definition works for infinite projective planes with the diagonal points of every quadrangle not collinear as well. Cherowitzo’s definition can be simplified (as we are not dealing with abstract ovals) to: whenever the point U is not on the oval and on a unique tangent line, and secant lines \(\ell ,m\) on U meet O at A, B and \(A',B'\), respectively, then the intersection of the tangent at A and the tangent at B, \(AA' \cap BB'\) and \(AB' \cap BA'\) are collinear as the definition of a harmonic oval in a Fano plane. We say an oval in a Fano plane is strongly harmonic whenever the point U is not on the oval, and secant lines \(\ell ,m\) on U meet O at A, B and \(A',B'\), respectively, then the intersection of the tangent at A and the tangent at B, \(AA' \cap BB'\) and \(AB' \cap BA'\) are collinear.

Theorem 5.6

A conic of a Pappian projective plane is a harmonic oval.

For finite planes, this is due to Ostrom in 1955 [72] for odd characteristic and to Cherowitzo in 1985 [17] for characteristic two, as well as Jungnickel and Vanstone in 1987 [24]. For infinite Pappian planes, Ostrom in 1981 [73, p. 190] remarks that most of the arguments of [72] work as well for the infinite case (and it turns out that this is one such to prove this theorem for characteristic not equal to two). For infinite Pappian planes of characteristic two, this theorem is due to Rigby in 1967 [43, 4.5 (iii)].

In order to prove analogues of the theorems of Mäurer, but without symmetry hypotheses, we need local versions of the harmonic condition.

In a projective plane with the diagonal points of every quadrangle not collinear, we say that an oval O with no three tangent lines concurrent is harmonic at U if U is an external point of O and whenever secant lines \(\ell ,m\) on U meet O at A, B and \(A',B'\), respectively, then the intersection of the tangent at A and the tangent at B, \(AA' \cap BB'\) and \(AB' \cap BA'\) are collinear.

In a projective plane with the diagonal points of every quadrangle collinear, we say that a central oval O is harmonic at U if U is a point not on O, not the nucleus of O, and whenever secant lines \(\ell ,m\) on U meet O at A, B and \(A',B'\), respectively, then the intersection of the tangent at A and the tangent at B, \(AA' \cap BB'\) and \(AB' \cap BA'\) are collinear.

In a projective plane with the diagonal points of every quadrangle not collinear, we say that an oval O with no three tangent lines concurrent is harmonic at \(\ell \) if \(\ell \) is secant to O at points A and B and O is harmonic at the intersection of the tangent at A and the tangent at B.

6 Harmonic ovals in Moufang Fano planes

Theorem 6.1

Let O be a central oval in a projective plane \(\pi \), t be a line on the nucleus of O, P be any point on t,  not on O, and not the nucleus of O. Suppose further that m is a line on P, not on N, \(m \cap O=\{A,B\}\), and that there is an elation g of \(\pi \) with centre P and axis t interchanging A and B. Then O is stabilised by g if and only if every quadrangle on O with P as a diagonal point has all its diagonal points on t.

Proof

Suppose every quadrangle on O with P as a diagonal point has all its diagonal points on t. Let O have nucleus N. Let C be a point of O, not on m and not on PN. Let \(PC \cap O=\{C,D\}\). Then \(\{A,B,C,D\}\) is a quadrangle on O with P as a diagonal point, and so all its diagonal points are on t. Hence \(E=AC \cap BD\) is on t, the axis of g. Now \(C^g\) is on PC, and C is on AE, so \(C^g\) is on \((AE)^g=BE=BD\). So \(C^g=PC \cap BD=D\). Moreover, g fixes every point on t, and so fixes \(PN \cap O\). Thus \(O^g=O\). Conversely, suppose O is stabilised by g. Let \(Q=\{A,B,C,D\}\) be a quadrangle on O with \(P=AB \cap CD\). Then \(A^g=B\) and \(C^g=D\). Now \((AC)^g=BD\), so \(X=AC \cap BD\) is fixed by g, and thus lies on the axis t of g. So all the diagonal points of Q are on t. \(\square \)

Corollary 6.2

Let O be a central oval in a translation plane with kernel of characteristic two, such that the translation line t is on the nucleus of O. Then Ois a translation oval with axis t if and only if every quadrangle on O with a diagonal point on t has diagonal line t.

There are many known translation ovals in non-Desarguesian translation planes of even order (see, e.g., the article of Jha-Johnson [93]).

Corollary 6.3

Let O be a central oval in a Moufang Fano plane, and t be a tangent line to O. Then Ois a translation oval with axis t if and only if every quadrangle on O with a diagonal point on t has diagonal line t.

Corollary 6.4

(Grüning [32], Satz 7) Let O be a central oval in a Pappian Fano plane \(\pi \). Then O is a conic if and only if every quadrangle on O has diagonal line on the nucleus of O (that is, if and only if O is strongly harmonic).

Corollary 6.5

(Penttila-Praeger [89], Theorem 4.3) Let O be an oval in a Pappian plane \(\pi \) of even order, and t be a tangent line to O. Then Ois a translation oval with axis t if and only if every quadrangle on O with a diagonal point on t has diagonal line t. In the affirmative case, O is equivalent to \(\{(1,x,x^{2^i}):x \in GF(2^h)\} \cup \{(0,0,1)\}\) for some i with \((i,h)=1\).

Proof

This follows from Theorem 4.15. \(\square \)

Corollary 6.6

(Grüning [32]) Let O be a central oval in a Moufang Fano plane \(\pi \). Then \(\pi \) is Pappian and O is a conic if and only if every quadrangle on O has diagonal line on the nucleus of O (that is, if and only if O is strongly harmonic).

Proof

This follows from Theorem 4.7. \(\square \)

Corollary 6.7

(Biondi [74]) Let O be an oval in a Pappian plane \(\pi \) of even order. Then O is a conic if and only if every quadrangle on O has diagonal line on the nucleus of O.

We quote from a paper from Ostrom of 1981 [73, pp. 191–192]: “So far as I know it is still an open question as to whether harmonic ovals ...exist that are not conics in Pappian planes.”

Corollary 6.8

Let O be a central oval in a Moufang Fano plane \(\pi \). Then O is a Moufang oval if and only if every quadrangle on O with a diagonal point on a tangent line has diagonal line a tangent line (that is, if and only if O is harmonic).

Proof

This follows from the definition of a Moufang oval. \(\square \)

So every Moufang oval is harmonic, and we have solved Ostrom’s open problem.

Corollary 6.9

Let O be a central oval in a Moufang Fano plane \(\pi \) such that there is a secant line t such that O is harmonic at every point U on t and not in O. Then \(\pi \) is Pappian and O is a conic.

Proof

This follows from Theorem 4.10. \(\square \)

7 Harmonic ovals in Moufang planes that are not Fano planes

Theorem 7.1

Let O be an oval in a projective plane \(\pi \) such that no three tangent lines are concurrent, t be a secant line to O, \(t \cap O=\{X, Y\}\), P be the intersection of the tangent to O at X and the tangent to O at Y. Suppose further that m is a line on P with \(m \cap O=\{A,B\}\), and that there is a homology g of \(\pi \) with centre P and axis t interchanging A and B. Then O is stabilised by g if and only if every quadrangle on O with P as a diagonal point has its other diagonal points on t.

Proof

Suppose every quadrangle on O with P as a diagonal point has its other diagonal points on t. Let C be a point of O, not on m. Let \(PC \cap O=\{C,D\}\). Then \(\{A,B,C,D\}\) is a quadrangle on O with P as a diagonal point, and so its other diagonal points are on t. Hence \(E=AC \cap BD\) is on t, the axis of g. Now \(C^g\) is on PC, and C is on AE, so \(C^g\) is on \((AE)^g=BE=BD\). So \(C^g=PC \cap BD=D\). Moreover, g fixes every point on t, and so fixes X and Y. Thus \(O^g=O\).

Conversely, suppose O is stabilised by g. Let \(Q=\{A,B,C,D\}\) be a quadrangle on O with \(P=AB \cap CD\). Then \(A^g=B\) and \(C^g=D\). Now \((AC)^g=BD\), so \(X=AC \cap BD\) is fixed by g, and thus lies on the axis t of g. So the other diagonal points of Q are on t. \(\square \)

Corollary 7.2

  • Let O be an oval in a Moufang plane \(\pi \) that is not a Fano plane such that no three tangent lines are concurrent. Then \(\pi \) is Pappian and O is a conic if and only if O is harmonic.

  • Let O be an oval in a Moufang plane \(\pi \) that is not a Fano plane, with no three tangent lines concurrent, such that there is a secant line t such that O is harmonic at every external point U on t and not in O. Then \(\pi \) is Pappian and O is a conic.

  • Let O be an oval in a Moufang plane \(\pi \) that is not a Fano plane, with no three tangent lines concurrent, such that there is a point A of O such that O is harmonic at every secant line t on A. Then \(\pi \) is Pappian and O is a conic.

  • Let O be an oval in a Moufang plane \(\pi \) that is not a Fano plane, with no three tangent lines concurrent, such that there is a tangent line t such that O is harmonic at every point U on t and not in O. Then \(\pi \) is Pappian and O is a conic.

  • Let O be an oval in a Moufang plane \(\pi \) that is not a Fano plane, with no three tangent lines concurrent, such that there is an external line t such that O is harmonic at every external point U on t. Then \(\pi \) is Pappian and O is a conic.

Proof

This follows from Theorems 4.8, 4.10, 4.11, 4.12, 4.13. \(\square \)

Examples of 2-transitive ovals In 1975, Kantor [46, Theorem 9] constructed ovals in the dual Lüneburg planes [54, 56] [55, Satz 10] of order \(q^2\) with nucleus the shears point, admitting Sz(q) acting 2-transitively which were further studied by Korchmáros in 1979 [49]. In 1965, Lüneburg [54, Chapter 14, Satz (14.10)] had proved a theorem on orbits of Sz(q) on the Lüneburg planes of order \(q^2\), building on work of Dembowski [21], in which a case had been open, corresponding to these ovals. By [46, remark after Theorem 9], with full details in [49], \(q-1\) isomorphism classes of such ovals are constructed. Since these ovals admit the Suzuki groups acting 2-transitively, we call them Suzuki ovals. In 2006, Maschietti [58], building on the odd order case dealt with by Korchmáros in 1978 [50] (and proved again with weaker hypotheses in Biliotti and Korchmáros in 1986 [62]), showed that these are the only ovals other than conics in finite projective planes which admit a 2-transitive group. These ovals are harmonic at every tangent line (since the group contains involutory elations with axis a tangent line).

Theorem 7.3

[58, Theorem 1.2] [62, Main Theorem] If an oval in a finite projective plane admits a collineation group acting 2-transitively, then either the plane is Pappian and the oval is a conic or the plane is the dual of the Lüneburg plane and the oval is equivalent to one of the Suzuki ovals.

Corollary 7.4

(Grüning [32], Satz 11) Let K be a set of five points, no three collinear in a Pappian Fano plane. Then the five diagonal lines of the quadrangles contained in K are concurrent.

Proof

K lies on a conic O. By Corollary 6.4, the diagonal line of every quadrangle on K lies on the nucleus of O. \(\square \)

It turns out that this theorem of Grüning of 1987 characterises Pappian Fano planes.

Theorem 7.5

(Grüning [32], Satz 11) If every set K of five points, no three collinear in a projective plane \(\pi \) has the property that the diagonal points of every quadrangle contained in K are collinear, and the five resulting lines are concurrent, then \(\pi \) is a Pappian Fano plane.

So if a set K of five points, no three collinear, in a Moufang Fano plane does not have all diagonal lines concurrent, then no oval containing K is a conic. This contrasts with the result that any two quadrangles of a Moufang plane are equivalent by a collineation ([33, 7.3.14]).

8 Ovals and degenerations of Pascal’s theorem

Theorem 8.1

Let O be a central oval in a Moufang Fano plane \(\pi \). Then \(\pi \) is Pappian and O is a conic if and only if every quadrangle ABCD on O has \(AB \cap CD\), \(AD \cap BC\) and the intersection of the tangent line to O at A and the tangent line to O at C collinear.

Proof

This is merely a rewriting of Corollary 6.6. \(\square \)

This is the four-point degeneration of Pascal’s theorem (applying to hexagons AABCCD) discovered as a property of real conics by Simson in 1735 [83, Libro V, Proposition VII]. There is another four-point degeneration of Pascal’s theorem (applying to hexagons AABBCD, and saying that the intersection of the tangent line to O at A and BC, \(AB \cap CD\), and the intersection of the tangent line to O at B and AD are collinear) discovered as a property of real conics by Ventimiglia in 1692 [3, 92].

It’s worth remarking that reading [17, 27, 28, 30, 38, 44, 69, 70, 72, 73] brings out the connections between the four-point degenerations of Pascal’s theorem and harmonic ovals, so that Theorem 8.1 is in no sense surprising.

We say that an oval O satisfies the four-point Pascal property if both four-point degenerations of Pascal’s theorem hold on O.

Theorem 8.2

Let O be an oval in a Moufang plane \(\pi \). Then \(\pi \) is Pappian and O is a conic if and only if O satisfies the four-point Pascal property.

Proof

If \(\pi \) is Pappian and O is a conic, then O satisfies the four-point Pascal property, by the Ventimiglia-Simson theorem. Conversely, suppose O satisfies the four-point Pascal property, By [44, 2.3], either O is central or no three tangent lines of O are concurrent. If O is central, then, by Theorem 8.1, \(\pi \) is Pappian and O is a conic. If no three tangent lines of O are concurrent, then by [69, Theorem 3.1, Theorem 4.1] and [44, 2.2] (or [8, Lemma 1]), O is harmonic. (We remark that Fernandes assumes that the plane is of odd order, but only uses that no three tangent lines are concurrent in proving Theorems 3.1 and 4.1, and that Rigby’s proof of his 2.2 and 2.3 only uses the four-point Pascal property.) Now, by Theorem 7.2, \(\pi \) is Pappian and O is a conic. \(\square \)

Corollary 8.3

(Pickert [34], Satz 3) Let O be an oval in a Pappian plane \(\pi \) that is not a Fano plane such that no three tangent lines are concurrent. Then the four-point degeneration of Pascal’s theorem holds on O if and only if O is a conic.

Pascal’s theorem goes back to 1640 as a property of real conics:

Theorem 8.4

(Pascal [75]) If a hexagon ABCDEF has vertices on an a conic of a Pappian projective plane, then \(AB \cap DE, BC \cap EF\) and \(CD \cap FA\) are collinear.

The joint converse of Pascal’s theorem and Pappus’ theorem was proved by Braikenridge in 1733 [12] and Maclaurin in 1735 [57] (but del Centina in 2020 [5] believes on indirect evidence that it was known to Pascal). Braikenridge and Maclaurin had a dispute over priority for this theorem (see Mills in 1984 [87]).

Theorem 8.5

(Braikenridge [12], Maclaurin [57]) In a Pappian projective plane, if three sides of a triangle pass through fixed points and two vertices vary along a line, then the other vertex traces a conic (which contains the fixed points), which may consist of a line pair.

The study of ovals via Pascal’s theorem began with the startling breakthrough of Buekenhout’s 1966 paper [28].

Theorem 8.6

(Buekenhout [28]) Let O be an oval in a projective plane. \(\pi \). Then \(\pi \) is Pappian and O is a conic if and only if O satisfies Pascal’s theorem.

The five-point degeneration of Pascal’s theorem (applying to hexagons AABCDE) was discovered as an affine property of real conics by l’Hôpital 1707 (p. 140, Corollaire IV, paragraph 208): if a pentagon ABCDE has vertices on an oval then the intersection of the tangent at A with CD, \(AB \cap DE\) and \(BC \cap EA\) are collinear. It was also discovered in this form, but not published by Newton c.1668 [68, Corollary, p. 191].

The three-point degeneration of Pascal’s theorem (applying to hexagons AABBCC) was discovered as a property of real conics by Maclaurin in 1735 [57].

Theorem 8.7

(Maclaurin [57]) If a triangle ABC has vertices on a conic then the intersection of the tangent at A with BC, the intersection of the tangent at B with CA and the intersection of the tangent at C with AB are collinear.

An improvement of Buekenhout’s theorem was obtained by Hofmannin 1971.

Theorem 8.8

(Hofmann [38]) Let O be an oval in a projective plane. \(\pi \). Then \(\pi \) is Pappian and O is a conic if and only if O satisfies the five-point degeneration of Pascal’s theorem, both four-point degenerations of Pascal’s theorem and the three-point degeneration of Pascal’s theorem.

Theorem 8.9

The three-point degeneration of Pascal’s theorem holds for every central oval O in a Fano plane \(\pi \).

Proof

Let N be the nucleus of O and ABC be a triangle with vertices on O. Then ABCN is a quadrangle with diagonal points collinear, since \(\pi \) is a Fano plane. Thus the intersection of the tangent at A with BC, the intersection of the tangent at B with CA and the intersection of the tangent at C with AB are collinear, since these are the diagonal points of ABCN. \(\square \)

For Pappian projective planes, a little more can be said. We first show that the four-point degeneration of Pascal’s theorem discovered by Ventimiglia characterises conics in Pappian planes.

Theorem 8.10

Let O be an oval in a Pappian plane. Then for every quadrangle ABCD with vertices on O, the intersection of the tangent line \(t_A\) to O at A and BC, \(AB \cap CD\), and the intersection of the tangent line \(t_B\) to O at B and AD are collinear if and only if O is a conic.

Proof

If O is a conic, this is a special case of the four point Pascal theorem for conics. Conversely, there is a unique conic K on A and tangent to \(t_A\), on Band tangent to \(t_B\) and on C. A point D such that ABCD is a quadrangle is on K if and only if \(t_A \cap BC\), \(AB \cap CD\) and \(t_B \cap AD\) are collinear, so every point D of O is on K. Since O is an oval, it follows that \(O=K\). \(\square \)

The dual of Pascal’s theorem was discovered for real conics by Brianchon in 1806.

Theorem 8.11

(Brianchon [13]) If a hexagon with sides abcdef has each side tangent to a conic of a Pappian projective plane, then the lines \((a \cap b)(d \cap e), (b cap c)(e \cap f)\) and \((c \cap d)(f \cap a)\) are concurrent.

For completeness, we mention that the four-line degeneration of Brianchon’s theorem dual to Ventimiglia’s four point Pascal was discovered for real conics by Newton [67, Corollary 2 to Lemma XXIV] in his Principia Mathematica in 1687, that the five-line degeneration of Brianchon’s theorem was discovered by Brianchon [13] (and also appeared in a later paper of 1817 [64]) and that the four-line degeneration of Brianchon’s theorem dual to Simson’s four point Pascal was discovered by Chasles in 1865 [65, Article 121, p. 89]. The three-line degeneration of Brianchon’s theorem was discovered for real conics by Ceva in 1678 [16].

Theorem 8.12

(Ceva [16]) If a triangle ABC has vertices on a conic in a Pappian plane that is not a Fano plane, then the triangle of tangents at A, B and C is in perspective with ABC from a point.

In the 2019 book of Kiss and Szőnyi, there is a characterisation of conics as ovals whose tangent lines form an oval in the dual plane and which satisfy the three-line degeneration of Brianchon’s theorem. In a personal communication of 22 May, 2021, Tamás Szőnyi told Tim Penttila that this proof is derived from the proof of Segre’s Theorem 3.10 in Section 2.13 of Kárteszi’s 1976 book [47], by noticing that Kárteszi’s proof, slightly modified, could be extended to work over all fields of characteristic not equal to two.

Theorem 8.13

(Kárteszi [35], Theorem 6.18) An oval in a Pappian plane that is not a Fano plane such that no three tangent lines are concurrent is a conic if it satisfies the three-line degeneration of Brianchon’s theorem.

We remark that Kárteszi in [35, Theorem 6.18] shows that an oval in a Pappian plane that is not a Fano plane whose tangent lines form an oval in the dual plane is a conic if it satisfies the three-line degeneration of Brianchon’s theorem: that every triangle on the oval is in perspective with the triangle of tangents at its vertices from a point. This can be deduced from Theorem 8.14, by applying Desargues’ theorem: every triangle on the oval is in perspective with the triangle of tangents at its vertices from a line, so the oval satisfies he three-point degeneration of Pascal’s theorem.

We now show that the three-point degeneration of Pascal’s theorem characterises conics in Pappian planes that are not Fano planes.

Theorem 8.14

(Pickert [34], Satz 4) Let O be an oval in a Pappian plane \(\pi \) that is not a Fano plane such that no three tangent lines are concurrent. Then the three-point degeneration of Pascal’s theorem holds on O if and only if O is a conic.

Proof

Consider a triangle on O and the triangle of tangents to O at its vertices. Since the three-point degeneration of Pascal’s theorem holds on O, these two triangles are in perspective from a line. By the Hilbert-Veblen-Young theorem, the plane is Desarguesian. By the dual of Desargues’ theorem, these two triangles are in perspective from a point. Hence, by Theorem 8.13, O is a conic. The converse is Maclaurin’s Theorem 8.7. \(\square \)

We can improve two of the preceding theorems, by using a consequence of a result of Tallini from 1995.

Theorem 8.15

(Tallini [88], Theorem 1.1) An oval in a Desarguesian plane that is not a Fano plane whose tangent lines form an oval in the dual plane is a conic in a Pappian plane if it satisfies the three-line degeneration of Brianchon’s theorem.

Theorem 8.16

Let O be an oval in a Desarguesian plane \(\pi \) that is not a Fano plane such that no three tangent lines are concurrent. Then the three-point degeneration of Pascal’s theorem holds on O if and only if \(\pi \) is Pappian and O is a conic.

Proof

Consider a triangle on O and the triangle of tangents to O at its vertices. Since the three-point degeneration of Pascal’s theorem holds on O, these two triangles are in perspective from a line. By the Hilbert-Veblen-Young theorem, the plane is Desarguesian. By the dual of Desargues’ theorem, these two triangles are in perspective from a point. Hence, by Theorem 8.15, \(\pi \) is Pappian and O is a conic. The converse is Maclaurin’s Theorem 8.7. \(\square \)

A line \(\ell \) is Pascalian w.r.t. an oval O if whenever ABCDEF is a hexagon inscribed in O with \(AB \cap DE\) on \(\ell \) and \(BC \cap EF\) on \(\ell \), it follows that \(CD \cap FA\) is on \(\ell \). Our final observation relates this concept to being harmonic for planes with the diagonal points of every quadrangle not collinear.

In [30, Section IV.3, p. 190], Faina observes that the ovals of Tits [40] have all tangent lines Pascalian. here, we sharpen that observation.

Theorem 8.17

Let O be a Moufang oval in a Moufang Fano plane \(\pi \). Then every tangent line of O is Pascalian. Conversely, if every tangent line of a central oval O in a Moufang Fano plane is Pascalian, then O is a Moufang oval.

Proof

Suppose t is a tangent line of O. By [28, Théorème 3], t is Pascalian if and only if the product of any three involutory elations with centre on t fixing O is involutory, noting that each involution of O is induced by an elation since O is a Moufang oval. But elations with the same axis commute, so this condition is satisfied, as these elations all have axis t. Conversely, O satisfies the four-point Pascal property w.r.t. tangent lines, so O is harmonic, and the result follows from Corollary 6.8. \(\square \)