Valyi pedal sequence

[Brendan McKay]

* David Wilson <davidwwilson at comcast.net> [050114 14:20]:
> Found the first six elements of Valyi's pedal sequence at
> http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Valyi.html
> and submitted as A102536.
> 
> Can someone find a reference?
> 
> Can the sequence be extended?


Here are some abstracts from MathSciNet.  I'll leave it for someone
else to visit a library and determine if any of them mention the
sequence defined in A102536.

Brendan.


MR0966233 (90b:11005)
Kingston, John G.(4-NOTT); Synge, John L.(IRL-DIAS)
The sequence of pedal triangles.
Amer. Math. Monthly 95 (1988), no. 7, 609--620.
11A07 (51M05)

The formulas that give the data of the pedal triangle $T'$ of a
given triangle $T$ depend on whether $T$ is acute or obtuse. Looking
at the sequence of pedal triangles $T,T',T"=(T')',\cdots,T^{(n)}$,
one cannot predict which triangles are acute and which obtuse.
The authors note that the angles of the $n$th pedal all satisfy
the same congruence $(\roman{mod}\,\pi)$. From this they obtain
that, for triangles $T$ with angles that are rational multiples
of $\pi$ with denominators $M=2^s(2^n±1)$ and such that no pedal
triangle degenerates, it follows that the sequence of triples
of angles is periodic. The length of the cycle is a divisor of
$\frac12\varphi(M_1)$ where $M_1$ is the largest odd divisor of $M$.
If the cycle length is $n$, then $2^{6n}T^{(6n)}$ is a translate of
$T$. Along the way, they prove for the Euler function of any odd
integer $\ge 3$ that $2^{\varphi(n)/2}\equiv ±1 (\roman{mod}\,n)$.



MR1976924 (2004d:60186)
Ding, Jiu(1-SMS); Hitt, L. Richard(1-SAL); Zhang, Xin-Min(1-SAL)
Markov chains and dynamic geometry of polygons. (English. English
summary)
Linear Algebra Appl. 367 (2003), 255--270.
60J10 (51M15 52B99 60D05)

The authors study the dynamics of some geometrically mildly natural
iterations on polygons embedded in the circle (an example: consider
complex numbers $z_1, \dots, z_n, z_{n+1} = z_1,$ with $|z_i| = 1$
for all $i.$ These correspond to the vertices of an $n$-gon $P_0$
inscribed in the unit circle. Now, construct a polygon $P_1,$ such
that the arguments of the $i$-th vertex of $P_1$ are the average of
the arguments of the $i$-th and the $i+1$st vertices of $P_0.$ Now
repeat to construct $P_2, \dots, P_n,\dots.$ The authors show that
in the limit a regular polygon is obtained).

The authors use simple linear algebra to derive their results.
This reviewer was surprised that the authors did not cite the
related and extremely interesting work of R. E. Schwartz \ref[see,
e.g., Experiment. Math. 10 (2001), no. 4, 519--528; MR1881752
(2003a:52020); Experiment. Math. 1 (1992), no. 1, 71--81; MR1181089
(93h:52002)].


MR1225542 (94m:51025)
Alexander, J. C.(1-MD)
The symbolic dynamics of the sequence of pedal triangles.
Math. Mag. 66 (1993), no. 3, 147--158.
51M04 (58F03)

This article revisits the topic of the behavior of the sequence of pedal
triangles \ref[see J. G. Kingston and J. L. Synge, Amer. Math. Monthly
95 (1988), no. 7, 609--620; MR0966233 (90b:11005); P. D. Lax, Amer.
Math. Monthly 97 (1990), no. 5, 377--381; MR1048909 (91h:51028); P.
Ungar, Amer. Math. Monthly 97 (1990), no. 10, 898--900; MR1079976
(92b:11055)] by using the symbolic dynamics method. This is a nice,
elementary introduction to the power and beauty of symbolic dynamics.


MR1079976 (92b:11055)
Ungar, Peter
Mixing property of the pedal mapping.
Amer. Math. Monthly 97 (1990), no. 10, 898--900.
11K99 (51M99)

A given triangle $T$ generates a sequence of triangles $T_n$, where
$T_{n+1}$ is the pedal triangle of $T_n$ $(n\ge0$, $T_0=T)$, this
being the triangle whose vertices are the feet of the altitudes of
$T_n$, and such a sequence is the pedal sequence of $T$. If $A$,
$B$, $C$ are the three angles of a triangle, the angles $A'$, $B'$,
$C'$ of its pedal triangle are simply related to $A$, $B$, $C$, and
the mapping $P\colon A, B,C\mapsto A',B',C'$ is called the pedal
mapping. Representing the shape of a given triangle as a point in
an equilateral triangle $E$, the barycentric coordinates of this
point being the angles of the triangle in units of $\pi$, \n J. G.
Kingston\en and \n J. L. Synge\en \ref[same journal 95 (1988), no.
7, 609--620; MR0966233 (90b:11005)] gave necessary and sufficient
conditions on the angles of $T$ for the three possibilities: (1)
$P^n$ is periodic; (2) the sequence stops by degeneration of the
triangle to a straight segment; (3) $P^n$ is infinite, without
periodicity. Referring to this article, \n P. D. Lax\en \ref[ibid.
97 (1990), no. 5, 377--381; MR1048909 (91h:51028)] showed the
ergodicity of $P$. In the article under review, a basic mixing
property of $P$ is proved, by a simple and elegant argument:
removing from the equilateral triangle $E$ a zero-measure set, the
author identifies each point $x$ of the remaining set $E'$ to a
base-4 sequence of digits $x_1x_2x_3\cdots$ in such a way that $Px$
is obtained by discarding the leftmost digit of $x$, and deduces the
result from Borel's theorem that almost all real numbers are normal.
For completeness, a short proof of this well-known result is given.
\{Reviewer's remark: There is a misprint in the formula of the last line
on p. 898, where $N(q,S,n)$ has to be replaced by $N(x,S,n)$.\}


MR1253798 (94j:51029)
Veldkamp, G. R.
Classical geometry. (Dutch)
Geometry, from art to science (Dutch), 1--15,
CWI Syllabi, 33,
Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1993.
51M04 (51-01 70B05)

The author gives some examples from a summer course on classical
geometry based on the studies of isometries. In the plane, he
starts with the theorem of Pompeiu: the distances of a point
from the vertices of an equilateral triangle are always the
sides of a triangle. Given the equilateral triangle, he then
asks for which points the Pompeiu triangle is acute, right,
or obtuse and discusses a few other problems of this type.
Next follows an introduction to plane kinematics that leads
to the Wallace line, a short study of pedal triangles, and an
introduction to configurations. Finally, there is an introduction
to three-dimensional isometries and kinematics. The survey also
contains an introduction to the literature and mentions as an
open challenge to find an elementary proof of the inequality
$$[(b\gamma-c\beta)^2+(c\alpha-a\gamma)^2+(a\beta-b\alpha)^2](a+b+c)
^{-2}<\pi^2/4.$$