## Hyperovals in the Known Projective Planes of Order 16

Incidence Geom. Zentralblatt MATH identifier Subjects Primary: 05B Difference sets number-theoretic, group-theoretic, etc. Keywords ovals polarity cyclic difference sets projective planes.

On polar ovals in cyclic projective planes. More by Philip P. In the finite Desarguesian planes PG 2, p n , the subplanes have orders which are the orders of the subfields of the finite field GF p n , that is, p i where i is a divisor of n.

In non-Desarguesian planes however, Bruck's theorem gives the only information about subplane orders. The case of equality in the inequality of this theorem is not known to occur. If such subplanes existed there would be projective planes of composite non-prime power order. A Fano subplane is a subplane isomorphic to PG 2,2 , the unique projective plane of order 2. If you consider a quadrangle a set of 4 points no three collinear in this plane, the points determine six of the lines of the plane. The remaining three points called the diagonal points of the quadrangle are the points where the lines that do not intersect at a point of the quadrangle meet.

The seventh line consists of all the diagonal points usually drawn as a circle or semicircle. The name Fano for this subplane is really a misnomer. Gino Fano — , in developing a new set of axioms for Euclidean geometry, took as an axiom that the diagonal points of any quadrangle are never collinear. This is called Fano's Axiom. A Fano subplane however violates Fano's Axiom.

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They really should be called Anti-Fano subplanes , but this name change has not had many supporters. In finite desarguesian planes, PG 2, q , Fano subplanes exist if and only if q is even that is, a power of 2. The situation in non-desarguesian planes is unsettled. They could exist in any non-desarguesian plane of order greater than 6, and indeed, they have been found in all non-desarguesian planes in which they have been looked for in both odd and even orders.

Projectivization of the Euclidean plane produced the real projective plane. The inverse operation — starting with a projective plane, remove one line and all the points incident with that line — produces an affine plane. More formally an affine plane consists of a set of lines and a set of points , and a relation between points and lines called incidence , having the following properties:.

### Projective Planes

The second condition means that there are parallel lines and is known as Playfair's axiom. The expression "does not meet" in this condition is shorthand for "there does not exist a point incident with both lines. The Euclidean plane and the Moulton plane are examples of infinite affine planes. A finite projective plane will produce a finite affine plane when one of its lines and the points on it are removed. The order of a finite affine plane is the number of points on any of its lines this will be the same number as the order of the projective plane from which it comes. The affine planes which arise from the projective planes PG 2, q are denoted by AG 2, q.

There is a projective plane of order N if and only if there is an affine plane of order N. When there is only one affine plane of order N there is only one projective plane of order N , but the converse is not true. The affine planes formed by the removal of different lines of the projective plane will be isomorphic if and only if the removed lines are in the same orbit of the collineation group of the projective plane.

These statements hold for infinite projective planes as well.

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The affine plane K 2 over K embeds into K P 2 via the map which sends affine non-homogeneous coordinates to homogeneous coordinates,. The complement of the image is the set of points of the form 0, x 1 , x 2. From the point of view of the embedding just given, these points are the points at infinity. They constitute a line in K P 2 — namely, the line arising from the plane. Consider for example the two lines. These lines have slope 0 and do not intersect.

They can be regarded as subsets of K P 2 via the embedding above, but these subsets are not lines in K P 2. Add the point 0, 1, 0 to each subset; that is, let. In fact, all lines in K 2 of slope 0, when projectivized in this manner, intersect at 0, 1, 0 in K P 2.

### Algebraic & Geometric Topology

The embedding of K 2 into K P 2 given above is not unique. Each embedding produces its own notion of points at infinity. For example, the embedding. When an affine plane does not have the form of K 2 with K a division ring, it can still be embedded in a projective plane, but the construction used above does not work. A commonly used method for carrying out the embedding in this case involves expanding the set of affine coordinates and working in a more general "algebra".

One can construct a coordinate "ring"—a so-called planar ternary ring not a genuine ring —corresponding to any projective plane. A planar ternary ring need not be a field or division ring, and there are many projective planes that are not constructed from a division ring.

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They are called non-Desarguesian projective planes and are an active area of research. The Cayley plane OP 2 , a projective plane over the octonions , is one of these because the octonions do not form a division ring. Conversely, given a planar ternary ring R,T , a projective plane can be constructed see below. The relationship is not one to one. A projective plane may be associated with several non-isomorphic planar ternary rings.

The ternary operator T can be used to produce two binary operators on the set R, by:. When the set of coordinates of a projective plane actually form a ring, a linear ternary operator may be defined in this way, using the ring operations on the right, to produce a planar ternary ring. Algebraic properties of this planar ternary coordinate ring turn out to correspond to geometric incidence properties of the plane.

For example, Desargues' theorem corresponds to the coordinate ring being obtained from a division ring , while Pappus's theorem corresponds to this ring being obtained from a commutative field. A projective plane satisfying Pappus's theorem universally is called a Pappian plane.

Alternative , not necessarily associative , division algebras like the octonions correspond to Moufang planes. There is no known purely geometric proof of the purely geometric statement that Desargues' theorem implies Pappus' theorem in a finite projective plane finite Desarguesian planes are Pappian. The converse is true in any projective plane and is provable geometrically, but finiteness is essential in this statement as there are infinite Desarguesian planes which are not Pappian. Degenerate planes do not fulfill the third condition in the definition of a projective plane.

They are not structurally complex enough to be interesting in their own right, but from time to time they arise as special cases in general arguments. They are:.

## [] The Class of Countable Projective Planes is Borel Complete

These seven cases are not independent, the fourth and fifth can be considered as special cases of the sixth, while the second and third are special cases of the fourth and fifth respectively. The special case of the seventh plane with no additional lines can be seen as an eighth plane. All the cases can therefore be organized into two families of degenerate planes as follows this representation is for finite degenerate planes, but may be extended to infinite ones in a natural way :.

The collection of fixed points and fixed lines of a collineation form a closed configuration , which is a system of points and lines that satisfy the first two but not necessarily the third condition in the definition of a projective plane. Thus, the fixed point and fixed line structure for any collineation either form a projective plane by themselves, or a degenerate plane. Collineations whose fixed structure forms a plane are called planar collineations. A homography or projective transformation of PG 2, K is a collineation of this type of projective plane which is a linear transformation of the underlying vector space.

Dempwolff and R. Some limited attempts have been made to construct new planes of order 16 by more general net replacement and other techniques, without success. It has been conjectured that this is in fact the complete list of planes of order I have made extensive use of Brendan McKay's celebrated software package nauty for computing graph automorphisms; also the computational algebra package GAP Graphs, Algorithms and Programming for some of the group computations e.

If you are aware of planes which I have overlooked in my list, I would appreciate an email message from you. For basic definitions and results on the subject of projective planes, please refer to P. Hughes and F. Following the table is a key to the table and a diagram showing connections between the planes.