OFFSET
1,1
COMMENTS
The first pedal of a triangle has as its vertices the feet of the perpendiculars of the original triangle. The (n+1)st pedal is the pedal of the n-th pedal.
From Fortschritte JFM 34.0551.02 on the Valyi paper: The triangle with corners the altitude bases of a given triangle ABC are called pedal triangles. The pedal triangle of this triangle is the second pedal triangle. Generally, we understand the n-th pedal triangle of the triangle ABC to be the pedal triangle of the (n-1)th pedal triangle. The author searches for and counts all triangles that are similar to their n-th pedal triangle, where all mutually similar triangle are counted as one.
The number of these is psi(n)=2^n(2^n-1). The number of triangles for which the n-th pedal triangle is the first that is similar to it is Sum_{d|n} mu(n/d) psi(d), where mu is the Möbius function. The author ends with a table of those triangles that are similar to their first, 2nd and 3rd pedal triangles.
Also, the number of 2 X n binary matrices that are "primitive"; that is, they cannot be expressed as a "tiling" by a smaller matrix; cf. A265627. - Jeffrey Shallit, Dec 11 2015
REFERENCES
Guilhem Gamard, Gwenaël Richomme, Jeffrey Shallit, Taylor J. Smith, Periodicity in rectangular arrays, Information Processing Letters 118 (2017) 58-63. See Table 1.
Hayashi, T. On the pedal triangles similar to the original triangles. Nieuw Archief (2) 10 (1912), 5-9. [Shows that there are 11 points whose pedal triangles are similar to the original triangle; those 11 points lie on a circle.]
de Vries, Jan, Über rechtwinklige Fusspunktdreiecke. Nieuw Archief (2) 9 (1910), 130-132. [The locus of those points that have rectangular pedal triangles with respect to a given triangle is determined by the three circles that cut the circumscribing circle orthogonally at two vertices of the triangle.]
Veldkamp, G. R. Classical geometry [Dutch], in Geometry, From Art to Science [Dutch], 1-15, CWI Syllabi, 33, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1993.
LINKS
J. C. Alexander, The symbolic dynamics of the sequence of pedal triangles, Math. Mag. 66 (1993), no. 3, 147-158.
Jiu Ding, L. Richard Hitt, Xin-Min Zhang, Markov chains and dynamic geometry of polygons, Linear Algebra Appl. 367 (2003), 255-270.
John G. Kingston, John L. Synge, The sequence of pedal triangles, Amer. Math. Monthly 95 (1988), no. 7, 609-620.
J. H. Smith, Gyula Valyi [Source of sequence.]
Peter Ungar, Mixing property of the pedal mapping, Amer. Math. Monthly 97 (1990), no. 10, 898-900.
J. Valyi, Über die Fusspunktdreiecke, Monatsh. f. Math. 14 (1903), 243-252.
CROSSREFS
KEYWORD
nonn
AUTHOR
David W. Wilson, Jan 13 2005
EXTENSIONS
Additional references supplied by Brendan McKay, Jan 14 2005
English summaries provided by Ralf Stephan, Jan 14 2005
More terms and formula from Valyi paper by Jeffrey Shallit, Nov 26 2015
STATUS
approved