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''circular plane, inversive plane''
 
''circular plane, inversive plane''
  
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1) any three distinct points are incident to one and only one circle;
 
1) any three distinct points are incident to one and only one circle;
  
2) given a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m0642901.png" /> on a circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m0642902.png" /> and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m0642903.png" /> not on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m0642904.png" />, there is a unique circle through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m0642905.png" /> whose common point with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m0642906.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m0642907.png" />;
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2) given a point $  A $
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on a circle $  \gamma $
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and a point $  B $
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not on $  \gamma $,  
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there is a unique circle through $  B $
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whose common point with $  \gamma $
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is $  A $;
  
 
3) there exist at least four distinct points not incident to a circle. Every circle is incident to at least three distinct points.
 
3) there exist at least four distinct points not incident to a circle. Every circle is incident to at least three distinct points.
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From a Möbius plane one can obtain an affine plane if one calls one of its points the ideal point, and calls the circles incident to this point straight lines.
 
From a Möbius plane one can obtain an affine plane if one calls one of its points the ideal point, and calls the circles incident to this point straight lines.
  
In three-dimensional projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m0642908.png" /> the points of an [[Ovoid(2)|ovoid]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m0642909.png" /> and the planes that intersect the ovoid at more than one point form, with the incidence relation inherited from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429010.png" />, a Möbius plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429011.png" /> (see [[#References|[1]]]). A Möbius plane is called egg-like if it is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429012.png" /> for some ovoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429013.png" />. Among the egg-like Möbius planes the best known is the model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429015.png" /> is the sphere in three-dimensional Euclidean space, i.e. the plane isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429017.png" /> is a non-ruled quadric in three-dimensional projective space over the field of real numbers.
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In three-dimensional projective space $  \mathbf P _ {3} $
 +
the points of an [[Ovoid(2)|ovoid]] $  o $
 +
and the planes that intersect the ovoid at more than one point form, with the incidence relation inherited from $  \mathbf P _ {3} $,  
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a Möbius plane $  M( o) $(
 +
see [[#References|[1]]]). A Möbius plane is called egg-like if it is isomorphic to $  M( o) $
 +
for some ovoid $  o $.  
 +
Among the egg-like Möbius planes the best known is the model $  M( S) $,  
 +
where $  S $
 +
is the sphere in three-dimensional Euclidean space, i.e. the plane isomorphic to $  M( c) $,  
 +
where $  c $
 +
is a non-ruled quadric in three-dimensional projective space over the field of real numbers.
  
A Möbius plane is said to be finite if it has a finite number of points and circles. Each circle in a Möbius plane has the same number of points, and through each point of the plane pass the same number of circles. By definition, the order of the plane is the number of points on a circle minus one. A Möbius plane of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429018.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429019.png" /> points and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429020.png" /> circles; through each point of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429021.png" /> circles pass. The following model of the Möbius plane of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429022.png" /> is the best known. The points of the plane are the elements of the [[Galois field|Galois field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429023.png" /> and the ideal point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429024.png" />; the circles of the plane are the images of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429025.png" /> under the group of permutations of the form
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A Möbius plane is said to be finite if it has a finite number of points and circles. Each circle in a Möbius plane has the same number of points, and through each point of the plane pass the same number of circles. By definition, the order of the plane is the number of points on a circle minus one. A Möbius plane of order $  n $
 +
contains $  n  ^ {2} + 1 $
 +
points and $  n( n  ^ {2} + 1) $
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circles; through each point of the plane $  n( n+ 1) $
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circles pass. The following model of the Möbius plane of order $  n = p  ^ {h} $
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is the best known. The points of the plane are the elements of the [[Galois field|Galois field]] $  \mathop{\rm GF} ( p  ^ {2h} ) $
 +
and the ideal point $  \{ \infty \} $;  
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the circles of the plane are the images of the set $  K = \mathop{\rm GF} ( p  ^ {h} ) \cup \{ \infty \} $
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under the group of permutations of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429026.png" /></td> </tr></table>
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$$
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x  \rightarrow 
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\frac{x  ^  \alpha  a+ c }{x  ^  \alpha  b+ d }
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,\ \
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a, b, c, d \in  \mathop{\rm GF} ( p  ^ {2h} ),\ \
 +
ad \neq bc,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429027.png" /></td> </tr></table>
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$$
 +
\alpha  \in  \mathop{\rm Aut}  \mathop{\rm GF} ( p  ^ {2h} ).
 +
$$
  
A necessary condition for the existence of a Möbius plane of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429028.png" /> is the existence of a finite affine plane of the same order. The uniqueness of the Möbius plane of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429029.png" /> has been proven [[#References|[5]]]. If a Möbius plane of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429030.png" /> contains a proper subplane of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064290/m06429033.png" /> (see [[#References|[2]]]).
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A necessary condition for the existence of a Möbius plane of order $  n $
 +
is the existence of a finite affine plane of the same order. The uniqueness of the Möbius plane of order $  n = 2, 3, 4, 5, 7, 11 $
 +
has been proven [[#References|[5]]]. If a Möbius plane of order $  n $
 +
contains a proper subplane of order m $,  
 +
then m \equiv n\ (  \mathop{\rm mod}  2) $
 +
and $  m  ^ {2} + m \leq  n $(
 +
see [[#References|[2]]]).
  
 
The classification of Möbius planes has been carried out (see [[#References|[3]]], [[#References|[4]]]). The planes are named after A. Möbius (1855), who laid the foundations of the theory of circles.
 
The classification of Möbius planes has been carried out (see [[#References|[3]]], [[#References|[4]]]). The planes are named after A. Möbius (1855), who laid the foundations of the theory of circles.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Dembowski,  "Finite geometries" , Springer  (1968)  pp. 254</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Dembowski,  D.R. Hughes,  "On finite inversive planes"  ''J. London Math. Soc.'' , '''40'''  (1965)  pp. 171–182</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C.H. Hering,  "Eine Klassifikation der Möbius-Ebenen"  ''Math. Z.'' , '''87'''  (1965)  pp. 252–262</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Krier,  "The Hering classification of Möbius planes" , ''Proc. Internat. Conf. Projective Planes'' , Washington State Univ. Press  (1973)  pp. 157–163</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.I. Istomina,  "Uniqueness of the circle plane of order 11"  ''Perm Univ. Uchen. Zap.'' , '''156'''  (1976)  pp. 81–83  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Dembowski,  "Finite geometries" , Springer  (1968)  pp. 254</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Dembowski,  D.R. Hughes,  "On finite inversive planes"  ''J. London Math. Soc.'' , '''40'''  (1965)  pp. 171–182</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C.H. Hering,  "Eine Klassifikation der Möbius-Ebenen"  ''Math. Z.'' , '''87'''  (1965)  pp. 252–262</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Krier,  "The Hering classification of Möbius planes" , ''Proc. Internat. Conf. Projective Planes'' , Washington State Univ. Press  (1973)  pp. 157–163</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.I. Istomina,  "Uniqueness of the circle plane of order 11"  ''Perm Univ. Uchen. Zap.'' , '''156'''  (1976)  pp. 81–83  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:02, 6 June 2020


circular plane, inversive plane

A plane whose elements form two disjoint sets: the set of points and the set of circles, and which is endowed with a symmetric incidence relation (relating points and circles). The incidence relation satisfies the following axioms:

1) any three distinct points are incident to one and only one circle;

2) given a point $ A $ on a circle $ \gamma $ and a point $ B $ not on $ \gamma $, there is a unique circle through $ B $ whose common point with $ \gamma $ is $ A $;

3) there exist at least four distinct points not incident to a circle. Every circle is incident to at least three distinct points.

From a Möbius plane one can obtain an affine plane if one calls one of its points the ideal point, and calls the circles incident to this point straight lines.

In three-dimensional projective space $ \mathbf P _ {3} $ the points of an ovoid $ o $ and the planes that intersect the ovoid at more than one point form, with the incidence relation inherited from $ \mathbf P _ {3} $, a Möbius plane $ M( o) $( see [1]). A Möbius plane is called egg-like if it is isomorphic to $ M( o) $ for some ovoid $ o $. Among the egg-like Möbius planes the best known is the model $ M( S) $, where $ S $ is the sphere in three-dimensional Euclidean space, i.e. the plane isomorphic to $ M( c) $, where $ c $ is a non-ruled quadric in three-dimensional projective space over the field of real numbers.

A Möbius plane is said to be finite if it has a finite number of points and circles. Each circle in a Möbius plane has the same number of points, and through each point of the plane pass the same number of circles. By definition, the order of the plane is the number of points on a circle minus one. A Möbius plane of order $ n $ contains $ n ^ {2} + 1 $ points and $ n( n ^ {2} + 1) $ circles; through each point of the plane $ n( n+ 1) $ circles pass. The following model of the Möbius plane of order $ n = p ^ {h} $ is the best known. The points of the plane are the elements of the Galois field $ \mathop{\rm GF} ( p ^ {2h} ) $ and the ideal point $ \{ \infty \} $; the circles of the plane are the images of the set $ K = \mathop{\rm GF} ( p ^ {h} ) \cup \{ \infty \} $ under the group of permutations of the form

$$ x \rightarrow \frac{x ^ \alpha a+ c }{x ^ \alpha b+ d } ,\ \ a, b, c, d \in \mathop{\rm GF} ( p ^ {2h} ),\ \ ad \neq bc, $$

$$ \alpha \in \mathop{\rm Aut} \mathop{\rm GF} ( p ^ {2h} ). $$

A necessary condition for the existence of a Möbius plane of order $ n $ is the existence of a finite affine plane of the same order. The uniqueness of the Möbius plane of order $ n = 2, 3, 4, 5, 7, 11 $ has been proven [5]. If a Möbius plane of order $ n $ contains a proper subplane of order $ m $, then $ m \equiv n\ ( \mathop{\rm mod} 2) $ and $ m ^ {2} + m \leq n $( see [2]).

The classification of Möbius planes has been carried out (see [3], [4]). The planes are named after A. Möbius (1855), who laid the foundations of the theory of circles.

References

[1] R. Dembowski, "Finite geometries" , Springer (1968) pp. 254
[2] P. Dembowski, D.R. Hughes, "On finite inversive planes" J. London Math. Soc. , 40 (1965) pp. 171–182
[3] C.H. Hering, "Eine Klassifikation der Möbius-Ebenen" Math. Z. , 87 (1965) pp. 252–262
[4] N. Krier, "The Hering classification of Möbius planes" , Proc. Internat. Conf. Projective Planes , Washington State Univ. Press (1973) pp. 157–163
[5] L.I. Istomina, "Uniqueness of the circle plane of order 11" Perm Univ. Uchen. Zap. , 156 (1976) pp. 81–83 (In Russian)

Comments

For more recent developments, see the chapters by I.M. Yaglom, J.F. Rigby, J.B. Wilker, and N.W. Johnson in [a1].

References

[a1] C. Davis (ed.) B. Grünbaum (ed.) F.A. Sherk (ed.) , The geometric vein (Coxeter-Festschrift) , Springer (1980)
How to Cite This Entry:
Möbius plane. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=M%C3%B6bius_plane&oldid=22815
This article was adapted from an original article by V.V. Afanas'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article