Translation plane

Every projective plane can be coordinatized by at least one planar ternary ring.[5] For translation planes, it is always possible to coordinatize with a quasifield.[6] However, some quasifields satisfy additional algebraic properties, and the corresponding planar ternary rings coordinatize translation planes which admit additional symmetries. Some of these special classes are:

• Nearfield planes - coordinatized by nearfields.
• Semifield planes - coordinatized by semifields, semifield planes have the property that their dual is also a translation plane.
• Moufang planes - coordinatized by alternative division rings, Moufang planes are exactly those translation planes that have at least two translation lines. Every finite Moufang plane is Desarguesian and every Desarguesian plane is a Moufang plane, but there are infinite Moufang planes that are not Desarguesian (such as the Cayley plane).

Given a quasifield with operations + (addition) and ${\displaystyle \cdot }$ (multiplication), one can define a planar ternary ring to create coordinates for a translation plane. However, it is more typical to create an affine plane directly from the quasifield by defining the points as pairs ${\displaystyle (a,b)}$ where ${\displaystyle a}$ and ${\displaystyle b}$ are elements of the quasifield, and the lines are the sets of points ${\displaystyle (x,y)}$ satisfying an equation of the form ${\displaystyle y=m\cdot x+b}$ , as ${\displaystyle m}$ and ${\displaystyle b}$ vary over the elements of the quasifield, together with the sets of points ${\displaystyle (x,y)}$ satisfying an equation of the form ${\displaystyle x=a}$ , as ${\displaystyle a}$ varies over the elements of the quasifield.[7]

Translation planes are related to spreads of odd-dimensional projective spaces by the André/Bruck-Bose construction.[8][9] A spread of PG(2n+1, K), where ${\displaystyle n\geq 1}$ is an integer and K a division ring, is a partition of the space into pairwise disjoint n-dimensional subspaces. In the finite case, a spread of PG(2n+1, q) is a set of qⁿ⁺¹ + 1 n-dimensional subspaces, with no two intersecting.

Given a spread S of PG(2n +1, K), the André/Bruck-Bose construction produces a translation plane as follows: Embed PG(2n+1, K) as a hyperplane ${\displaystyle \Sigma }$ of PG(2n+2, K). Define an incidence structure A(S) with "points," the points of PG(2n+2, K) not on ${\displaystyle \Sigma }$ and "lines" the (n+1)-dimensional subspaces of PG(2n+2, K) meeting ${\displaystyle \Sigma }$ in an element of S. Then A(S) is an affine translation plane. In the finite case, this procedure produces a translation plane of order qⁿ⁺¹.

The converse of this statement is almost always true.[10] Any translation plane which is coordinatized by a quasifield that is finite-dimensional over its kernel K (K is necessarily a division ring) can be generated from a spread of PG(2n+1, K) using the André/Bruck-Bose construction, where (n+1) is the dimension of the quasifield, considered as a module over its kernel. An instant corollary of this result is that every finite translation plane can be obtained from this construction.

Let ${\displaystyle \Sigma }$ be the projective space PG(2n+1, K) for ${\displaystyle n\geq 1}$ an integer, and K a division ring. A regulus[11] R in ${\displaystyle \Sigma }$ is a collection of pairwise disjoint n-dimensional subspaces with the following properties:

1. R contains at least 3 elements
2. Every line meeting three elements of R, called a transversal, meets every element of R
3. Every point of a transversal to R lies on some element of R

Any three pairwise disjoint n-dimensional subspaces in ${\displaystyle \Sigma }$ lie in a unique regulus.[12] A spread S of ${\displaystyle \Sigma }$ is regular if for any three distinct n-dimensional subspaces of S, all the members of the unique regulus determined by them are contained in S. For any division ring K with more than 2 elements, if a spread S of PG(2n+1, K) is regular, then the translation plane created by that spread via the André/Bruck-Bose construction is a Moufang plane. A slightly weaker converse holds: if a translation plane is Pappian, then it can be generated via the André/Bruck-Bose construction from a regular spread.[13]

In the finite case, K must be a field of order ${\displaystyle q>2}$, and the classes of Moufang, Desarguesian and Pappian planes are all identical, so this theorem can be refined to state that a spread S of PG(2n+1, q) is regular if and only if the translation plane created by that spread via the André/Bruck-Bose construction is Desarguesian.

All spreads of PG(2n+1, 2) are trivially regular, since a regulus only contains three elements. While the only translation plane of order 8 is Desarguesian, there are known to be non-Desarguesian translation planes of order 2^e for every integer ${\displaystyle e\geq 4}$.[14]

• Hall planes
• André planes
• Subregular planes
• Nearfield planes
• Semifield planes

It is well known that the only projective planes of order 8 or less are Desarguesian, and there are no known non-Desarguesian planes of prime order.[15] Finite translation planes must have prime power order. There are four projective planes of order 9, of which two are translation planes: the Desarguesian plane, and the Hall plane. The following table details the current state of knowledge:

An algebraic representation of (affine) translation planes can be obtained as follows: Let V be a 2n-dimensional vector space over a field F. A spread of V is a set S of n-dimensional subspaces of V that partition the non-zero vectors of V. The members of S are called the components of the spread and if V_i and V_j are distinct components then V_i ⊕ V_j = V. Let A be the incidence structure whose points are the vectors of V and whose lines are the cosets of components, that is, sets of the form v + U where v is a vector of V and U is a component of the spread S. Then:[27]

A is an affine plane and the group of translations x → x + w for w in V is an automorphism group acting regularly on the points of this plane.

A regular spread may be constructed in the following way. Let F be a field and E an n-dimensional extension field of F. Let V = E² considered as a 2n-dimensional vector space over F. The set of all 1-dimensional subspaces of V over E (and hence, n-dimensional over F) is a regular spread of V.

In the finite case, the field E = GF(qⁿ) can be represented as a subring of the n × n matrices over F = GF(q). With respect to a fixed basis of E over F, the multiplication maps, x → αx for α in E, are F-linear transformations and can be represented by n × n matrices over F. These matrices are the spread matrices of a regular spread.[28]

As a specific example, the following nine matrices represent GF(9) as 2 × 2 matrices over GF(3) and so provide a spread set of AG(2, 9).

${\displaystyle \left[{\begin{matrix}0&0\\0&0\end{matrix}}\right],\left[{\begin{matrix}1&0\\0&1\end{matrix}}\right],\left[{\begin{matrix}2&0\\0&2\end{matrix}}\right],\left[{\begin{matrix}0&1\\2&0\end{matrix}}\right],\left[{\begin{matrix}1&1\\2&1\end{matrix}}\right],\left[{\begin{matrix}2&1\\2&2\end{matrix}}\right],\left[{\begin{matrix}0&2\\1&0\end{matrix}}\right],\left[{\begin{matrix}1&2\\1&1\end{matrix}}\right],\left[{\begin{matrix}2&1\\2&2\end{matrix}}\right].}$

The set of transversals of a regulus R also form a regulus, called the opposite regulus of R. If a spread S of PG(3, q) contains a regulus R, the removal of R and replacing it by its opposite regulus produces a new spread S^∗. This process is a special case of a more general process called derivation or net replacement.[29]

Starting with a regular spread of PG(3, q) and deriving with respect to any regulus produces a Hall plane. In more generality, the process can be applied independently to any collection of reguli in a regular spread, yielding a subregular spread;[30] the resulting translation plane is called a subregular plane. The André planes form a special subclass of subregular planes, of which the Hall planes are the simplest examples.

• André, Johannes (1954), "Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe", Mathematische Zeitschrift, 60: 156–186, doi:10.1007/BF01187370, ISSN 0025-5874, MR 0063056, S2CID 123661471
• Ball, Simeon; John Bamberg; Michel Lavrauw; Tim Penttila (2003-09-15), Symplectic Spreads (PDF), Polytechnic University of Catalonia, retrieved 2008-10-08
• Bruck, R.H. (1969), R.C.Bose and T.A. Dowling (ed.), "Construction Problems of finite projective planes", Combinatorial Mathematics and Its Applications, Univ. of North Carolina Press, pp. 426–514
• Bruck, R. H.; Bose, R. C. (1966), "Linear Representations of Projective Planes in Projective Spaces" (PDF), Journal of Algebra, 4: 117–172, doi:10.1016/0021-8693(66)90054-8
• Bruck, R. H.; Bose, R. C. (1964), "The Construction of Translation Planes from Projective Spaces" (PDF), Journal of Algebra, 1: 85–102, doi:10.1016/0021-8693(64)90010-9
• Czerwinski, Terry; Oakden, David (1992). "The translation planes of order twenty-five". Journal of Combinatorial Theory, Series A. 59 (2): 193–217. doi:10.1016/0097-3165(92)90065-3.
• Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, MR 0233275
• Dempwolff, U. (1994). "Translation planes of order 27". Designs, Codes and Cryptography. 4 (2): 105–121. doi:10.1007/BF01578865. ISSN 0925-1022. S2CID 12524473.
• Dover, Jeremy M. (2019-02-27). "A genealogy of the translation planes of order 25". arXiv:1902.07838 [math.CO].
• Hall, Marshall (1943), "Projective planes" (PDF), Trans. Amer. Math. Soc., 54 (2): 229–277, doi:10.2307/1990331, JSTOR 1990331
• Hughes, Daniel R.; Piper, Fred C. (1973), Projective Planes, Springer-Verlag, ISBN 0-387-90044-6
• Johnson, Norman L.; Jha, Vikram; Biliotti, Mauro (2007), Handbook of Finite Translation Planes, Chapman&Hall/CRC, ISBN 978-1-58488-605-1
• Knuth, Donald E. (1965), "A Class of Projective Planes" (PDF), Transactions of the American Mathematical Society, 115: 541–549, doi:10.2307/1994285, JSTOR 1994285
• Lüneburg, Heinz (1980), Translation Planes, Berlin: Springer Verlag, ISBN 0-387-09614-0
• Mathon, Rudolf; Royle, Gordon F. (1995). "The translation planes of order 49". Designs, Codes and Cryptography. 5 (1): 57–72. doi:10.1007/BF01388504. ISSN 0925-1022. S2CID 1925628.
• McKay, Brendan D.; Royle, Gordon F. (2014). "There are 2834 spreads of lines in PG(3,8)". arXiv:1404.1643 [math.CO].
• Moorhouse, Eric (2007), Incidence Geometry (PDF), archived from the original (PDF) on 2013-10-29
• Reifart, Arthur (1984). "The classification of the translation planes of order 16, II". Geometriae Dedicata. 17 (1). doi:10.1007/BF00181513. ISSN 0046-5755. S2CID 121935740.
• Sherk, F. A.; Pabst, Günther (1977), "Indicator sets, reguli, and a new class of spreads" (PDF), Canadian Journal of Mathematics, 29 (1): 132–54, doi:10.4153/CJM-1977-013-6

• Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) Foundations of Translation Planes, Marcel Dekker ISBN 0-8247-0609-9 .

• Lecture Notes on Projective Geometry
• Publications of Keith Mellinger