triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a Hofstadter point is a special point associated with every

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * Planes (gen ...

. In fact there are several Hofstadter points associated with a triangle. All of them are

triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For example ...

s. Two of them, the Hofstadter zero-point and Hofstadter one-point, are particularly interesting. They are two transcendental triangle centers. Hofstadter zero-point is the center designated as X(360) and the Hofstafter one-point is the center denoted as X(359) in

Clark Kimberling Clark Kimberling (born November 7, 1942 in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer seq ...

Encyclopedia of Triangle Centers The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville. , the l ...

. The Hofstadter zero-point was discovered by

Douglas Hofstadter Douglas Richard Hofstadter (born February 15, 1945) is an American scholar of cognitive science, physics, and comparative literature whose research includes concepts such as the sense of self in relation to the external world, consciousness, an ...

in 1992.

Hofstadter triangles

Let ''ABC'' be a given triangle. Let ''r'' be a positive real constant. Rotate the line segment ''BC'' about ''B'' through an angle ''rB'' towards ''A'' and let ''L_BC'' be the line containing this line segment. Next rotate the line segment ''BC'' about ''C'' through an angle ''rC'' towards ''A''. Let ''L'_BC '' be the line containing this line segment. Let the lines ''L_BC'' and ''L'_BC '' intersect at ''A''(''r''). In a similar way the points ''B''(''r'') and ''C''(''r'') are constructed. The triangle whose vertices are ''A''(''r''), ''B''(''r''), ''C''(''r'') is the Hofstadter ''r''-triangle (or, the ''r''-Hofstadter triangle) of triangle ''ABC''.

Special case

*The Hofstadter 1/3-triangle of triangle ''ABC'' is the first Morley's triangle of triangle ''ABC''. Morley's triangle is always an

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...

. *The Hofstadter 1/2-triangle is simply the incentre of the triangle.

Trilinear coordinates of the vertices of Hofstadter triangles

The

trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...

of the vertices of the Hofstadter ''r''-triangle are given below: :''A''(''r'') = ( 1 , sin ''rB'' / sin (1 − ''r'')''B'' , sin ''rC'' / sin (1 − ''r'')''C'' ) :''B''(''r'') = ( sin ''rA'' / sin (1 − ''r'')''A'' , 1 , sin ''rC'' / sin (1 − ''r'')''C'' ) :''C''(''r'') = ( sin ''rA'' / sin (1 − ''r'')''A'' , sin (1 − ''r'')''B'' / sin ''rB'' , 1 )

Hofstadter points

For a positive real constant ''r'' > 0, let ''A''(''r'') ''B''(''r'') ''C''(''r'') be the Hofstadter ''r''-triangle of triangle ''ABC''. Then the lines ''AA''(''r''), ''BB''(''r''), ''CC''(''r'') are concurrent. The point of concurrence is the Hofstdter ''r''-point of triangle ''ABC''.

Trilinear coordinates of Hofstadter ''r''-point

The trilinear coordinates of Hofstadter ''r''-point are given below. :( sin ''rA'' / sin ( ''A'' − ''rA'') , sin ''rB'' / sin ( ''B − ''rB'' ) , sin ''rC'' / sin ( ''C'' − ''rC'') )

Hofstadter zero- and one-points

The trilinear coordinates of these points cannot be obtained by plugging in the values 0 and 1 for ''r'' in the expressions for the trilinear coordinates for the Hofstdter ''r''-point. :Hofstadter zero-point is the limit of the Hofstadter ''r''-point as ''r'' approaches zero. :Hofstadter one-point is the limit of the Hofstadter ''r''-point as ''r'' approaches one. Trilinear coordinates of Hofstadter zero-point : = lim _{''r'' → 0} ( sin ''rA'' / sin ( ''A'' − ''rA'') , sin ''rB'' / sin ( ''B'' − ''rB'' ) , sin ''rC'' / sin ( ''C'' − ''rC'') ) : = lim _{''r'' → 0} ( sin ''rA'' / ''r'' sin ( ''A'' − ''rA'') , sin ''rB'' / ''r'' sin ( ''B'' − ''rB'' ) , sin ''rC'' / ''r'' sin ( ''C'' − ''rC'') ) : = lim _{''r'' → 0} ( ''A'' sin ''rA'' / ''rA'' sin ( ''A'' − ''rA'') , ''B'' sin ''rB'' / ''rB'' sin ( ''B'' − ''rB'' ) , ''C'' sin ''rC'' / ''rC'' sin ( ''C'' − ''rC'') ) : = ( ''A'' / sin ''A'' , ''B'' / sin ''B'' , ''C'' / sin ''C'' ) ), as lim _{''r'' → 0} sin ''rA'' / ''rA'' = 1, etc. : = ( ''A'' / ''a'', ''B'' / ''b'', ''C'' / ''c'' ) Trilinear coordinates of Hofstadter one-point : = lim _{''r'' → 1} ( sin ''rA'' / sin ( ''A'' − ''rA'') , sin ''rB'' / sin ( ''B'' − ''rB'' ) , sin ''rC'' / sin ( ''C'' − ''rC'') ) : = lim _{''r'' → 1} ( ( 1 − ''r'' ) sin ''rA'' / sin ( ''A'' − ''rA'') , ( 1 - ''r'' ) sin ''rB'' / sin ( ''B'' − ''rB'' ) , ( 1 − ''r'' )sin ''rC'' / sin ( ''C'' − ''rC'') ) : = lim _{''r'' → 1} ( ( 1 − ''r'' ) ''A'' sin ''rA'' / ''A'' sin ( ''A'' − ''rA'') , ( 1 − ''r'' ) ''B'' sin ''rB'' / ''B'' sin ( ''B'' − ''rB'' ) , ( 1 − ''r'' ) ''C'' sin ''rC'' / ''C'' sin ( ''C'' − ''rC'') ) : = ( sin ''A'' / ''A'' , sin ''B'' / ''B'' , sin ''C'' / ''C'' ) ) as lim _{''r'' → 1} ( 1 − ''r'' ) ''A'' / sin ( ''A'' − ''rA'' ) = 1, etc. : = ( ''a'' / ''A'', ''b'' / ''B'', ''c'' / ''C'' )

References

{{Douglas Hofstadter Triangle centers