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Geometry and the Discrete Fourier Transform

We'll show a relationship between some elementary geometry of the plane and the Discrete Fourier Transform, that is a starting point for ramifications for polynomials, circulant matrices and interpolation, and even sculpture. We employ some recent technology to do this. ...

Patrick D. F. Ion
Mathematical Reviews
ion at ams.org

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Introduction

Many of the most curious and popular theorems of plane geometry are those of Coincidence, Collinearity, and Equilateralness (also sometimes called Equilaterality): a certain collection of lines all intersect in a single point, or a collection of points in the plane lies on a line, or a certain triple of points forms an equilateral triangle.

The Discrete Fourier Transform is a tool used to efficiently and accurately approximate the Fourier Transform. It is applied in electrical engineering to analyze a signal into its components at various frequencies, and in mechanical engineering to analyze vibrations of a system.

There is a surprising connection between these two topics.

The starting point is to consider points in the plane as complex numbers $z = x+\ii y\in\cplx.$

Complex numbers as points

To see a note on new technology behind this document, and explanation of the buttons above, click on the following up-arrow; to hide it again click on the corresponding down-arrow when it is shown .

This pairing of up- and down-arrows is used to Show or Hide asides or details of equations throughout this month's column.

Part I. The theorems

Theorems of Coincidence: Triangle Medians

The median lines of a triangle are the lines drawn from each vertex to the midpoint of the opposite side. The simplest Coincidence Theorem asserts that these three lines meet in a single point (which happens to be the center of gravity or centroid of the triangle).

This triangle has vertices $A_0$, $A_1$ and $A_2$, and the points $M_0$, $M_1$ and $M_2$ are the midpoints of opposite sides; $A_0M_0$, $A_1M_1$ and $A_2M_2$ are the medians. In the interactive pop-up the blue points $A_0$, $A_1$ and $A_2$ can be dragged about with the mouse to change the triangle. The red point $G$ is the triangle's centroid.

Triangle Medians and the Centroid

There are other well known examples: the angle bisectors all intersect at the point $I$, the center of the inscribed circle (called the incenter), the perpendicular bisectors of the sides intersect at the circumcenter $C$, the altitudes intersect at the orthocenter $O$.

Collinearity theorems

The prime example of a collinearity theorem is that the circumcenter $C$, the centroid $G$ and the orthocenter $O$ are collinear. Below the altitudes are in black, medians are in green and perpendicular bisectors in blue. The line through their coincidence is shown in orange.

The Euler Line

Equilateralness - Napoleon's Theorem

Napoleon's Theorem, though probably not directly due to the emperor himself, has long borne his name.

Its simplest form says that if one constructs an equilateral triangle external to each side of a nondegenerate plane triangle, and takes the three centroids of the resulting triangles, then these points form an equilateral triangle. A similar construction using internal triangles also leads to another equilateral triangle.

Napoleon Triangles

The most famous equilaterality theorem is Morley's Miracle from about 1901 which states that the pairwise intersections of adjacent internal angle trisectors are the vertices of an equilateral triangle.

A pentagon construction

A relatively recent construction was discovered by the Fields Medalist Jesse Douglas. It was devised probably around 1940, but only published in 1960, in a piece written in honor of a colleague Jekuthiel Ginsburg. He specifies a fixed construction that from any pentagon derives an associated convex pentagon and pentagram.

Consider a pentagon in the plane. Suppose the five vertex points are $ p_0, p_1, p_3 p_3, p_4$, taken to be numbers in $\cplx$. So we are given a complex vector in $\cplx^5$, which is to say an array of complex numbers $$ p=(p_0,p_1,p_2,p_3,p_4).$$ No one point of the five is to be distinguished. What is important is the cyclic order, so we shall consider the indices $i$ of $p_i$ modulo $5$. Sums in the index are cyclic, so that, for instance, $p_{3+4} = p_2$.

Douglas tells us: Join each vertex $p_i$ to the midpoint of the edge opposite between the points $p_{i+2}$ and $p_{i+3}$; then extend that line by an extra $(1/\sqrt{5})$ of its length to create a new vertex $q_i$. The new pentagon $(q_0,q_1, q_2, q_3, q_4)$ is always the affine image of a convex regular pentagon. Also, if the lines to the opposite midpoints are shortened by a similar decrement of $(1/\sqrt{5})$ to give new vertices $r_i$, then $(r_0, r_1, r_2, r_3, r_4)$ is the affine image of a regular star pentagon, commonly called a pentagram.

We plot a chosen example of $p=(p_0,p_1,p_2,p_3,p_4)$ but, as usual, the vertices can be dragged about in the interactive pop-up figures.

A pentagon example

We see $$ q_i = p_i + \left(1+{\textstyle\sfrac1{\sqrt{5}}}\right) \left({{\textstyle\sfrac12}}(p_{i+2}+p_{i+3}) - p_i\right) , $$ outlined in green in the figure.

A pentagon example p and a constructed pentagon q

and similarly $$ r_i = p_i + \left(1-{\textstyle\sfrac11{\sqrt{5}}}\right)\left({\textstyle\sfrac12}(p_{i+2}+p_{i+3}) - p_i\right) , $$ outlined in red.

A pentagon example p and a constructed pentagon r

One may put all this in a diagram with evrything drawn, which it is amusing to pull about to see the effect of the construction.

We see that starting from an arbitrary pentagon, a list of five points, we get by a fixed construction two new pentagons, which are, respectively, an affine image of a standard convex pentagon and an affine image of a standard pentagram.

Aside: affine transformations

 

Part II. Geometry with the Discrete Fourier Transform

The Discrete Fourier Transform

A polygon with $N$ sides is a sequence of vertices $p_{0}$, $p_{1}$, ... , $p_{N-1}$. None of the vertices is particularly more important than the others, so all that really matters is the cyclic order. In other words, what we have at hand is a complex-valued function $p_{j}$ on the cyclic group $\intg/N$ (short for $\intg/N\intg$.

The vector space $\cplx[\intg/N]$ of all complex-valued functions on $\intg/N$ has dimension $N$, and the natural coordinate system on this space assigns to a function $f$ the array $(f_{0}, \dots , f_{N-1})$ in $\cplx^{N}$. But there is another useful coordinate system as well. To tell you what this is, we need to specify a new basis of the vector space. Let $\omega_{N} = \ee^{2 \pi \ii/N}$, and for each $a$ in $\intg$, let $\chi_{a}$ (more precisely $\chi_{N,a}$) be the function taking $j$ to $\omega_{N}^{aj}$. This is a function on $\intg/N$, since $\omega_{N}^{N} = 1$, that is $\omega_{N}$ is an $N$-th root of unity. Furthermore, $\chi_{a}$ only depends on $a$ modulo $N$, so there are $N$ of these functions.

We assign to the complex vector space of complex-valued functions on $\intg/N$ a a Hermitian inner product: $$ p \Dot q = { 1 \over N} \sum_{j=0}^{N-1} p_{j}\bar{q}_{j} = (1/N)(p_{0}\bar{q}_{0} + \cdots + p_{N-1}\bar{q}_{N-1}) . $$ The function $\chi_{0}$ is just the constant function taking all $j$ to $1$. We have $$ \chi_{a} \Dot \chi_{0} = \begin{cases} 1 \text{ if } a=0 \cr 0 \text{ otherwise} \cr \end{cases} $$ and hence $$ \chi_{a} \Dot \chi_{b} = \begin{cases} 1 \text{ if } a=b \cr 0 \text{ otherwise.}\cr \end{cases} $$ This means that the functions $\chi_{a}$ for $a$ in $\intg/N$ are an orthonormal basis for the space of functions on $\intg/N$. If $f$ is any such function, we may therefore write $$ f = \sum_{a=0}^{N-1} \hat{f}_{a} \chi_{a} \quad \text{ where } \quad \hat{f}_{a} = f \Dot \chi_{a} $$ In particular, $\hat f_{0}$ is the center of gravity (a.k.a. centroid, barycenter or mean) of the values of $f$, $$ \hat{f}_{0} = \frac{1}{N} (f_0 + \cdots + f_{N-1}) $$ and $\hat{f}_{0},\dots,\hat{f}_{N-1}$ are the coefficients in the expression of the arbitrary function $f$, an $N$-gon, as a linear sum of standard ones.

For example, the function $f_{j} = \chi_{1}(j)$ lists the vertices of the symmetric regular polygon of $N$ sides inscribed in the unit circle. Similarly $g_{j} = \chi_{N-1}(j)$ gives the standard negatively oriented unit $N$-gon. The combination $\lambda \chi_{1} +\mu \chi_{N-1}$ with $\lambda$, $\mu$ in $\cplx$ then gives a general affine convex $N$-gon.

Triangle geometry

The claim is that much classical plane geometry has a natural expression from the point of view outlined above. Start by looking at the standard types of triangles. For that we'll take $\omega := \omega_3^{\strut} = \frac12(-1 + \ii \sqrt3)$ until further notice. The standard triangles are $\chi_0=(1,1,1)$, the degenerate one with 3 points at 1, the unit equilateral triangle $\chi_1=(1,\omega,\omega^2)$, and its reverse $\chi_2=(1,\omega^2,\omega)$.

The picture below shows the simple average of a standard equilateral triangle and its reverse, namely $\sfrac12(\chi_1+\chi_2)$. It is a digon (a segment) with a double point at the left-hand end.

Triangle and anti-triangle averaged

We consider only triangles whose centroids are at the origin; after all, it is easy enough to move any triangle into this simplifying position by a translation, and all interesting theorems of plane geometry are invariant under translations.

The fully general case is then the addition of two arbitrarily sized and rotated standard triangles. This is a case where the interactive form of the diagram can be quite helpful.

The triangle $C$ is the average of triangles $A$ and $B$ so $C_i$ is midway between $A_i$ abd $B_i$. That is, $C$ is the triangle ${\scriptstyle\frac12}(\lambda \ee^{\ii \alpha}\chi_1 +\mu \ee^{\ii \beta}\chi_2)$.

The red sliders are $\lambda$ controlling the size of the red image of the standard triangle and $\alpha$ controlling its angle of rotation. Then the blue $\mu$ and $\beta$ are the corresponding controls for the blue image of the reversed standard triangle.

General scaled and rotated triangle and anti-triangle averaged

The general sum of this form, with six real parameters $\lambda,\mu,\nu,\alpha,\beta,\gamma\in \reals$, is $$ \lambda \ee^{\ii \alpha}\chi_1 +\mu \ee^{\ii \beta}\chi_2 + \nu \ee^{\ii \gamma}\chi_0 $$ and can represent any triangle in the plane; the centre of gravity is $\nu \ee^{\ii \gamma}\chi_0$.

One can spend quite some time playing with the interactive figures for polygon superpositions.

Proof of Median Coincidence

If the vertices of a triangle are $z=(z_0,z_1,z_2)$, the midpoints of the sides are $$ $$ where $I$ is the identity and $J$ is the cyclic permutation $$ J: (z_0,z_1,z_2) \mapsto (z_1,z_2,z_0) . $$ The points on the line from $z_0$ to $\sfrac12(z_2+z_1)$ are of the form, for some $\lambda \in \reals$ $$ (1-\lambda)z_0 + \lambda {\textstyle\sfrac12}(z_2+z_1) $$ and similarly on the line from $z_1$ to $\sfrac12(z_0+z_2)$ we have, for some $\mu \in \reals$ $$ (1-\mu)z_1 + \mu {\textstyle\sfrac12}(z_0+z_2). $$ The only solution for a point that lies on both lines is $$ (1-{\textstyle\sfrac23})z_1 + {\textstyle\sfrac23}{\textstyle\sfrac12}(z_0+z_2) = {\textstyle\sfrac13}(z_0 + z_1 + z_2) = \chi_0 \Dot z. $$ The common intersection point of medians as the centre of gravity has become obvious.▓

Proof of other coincidences and collinearity

Other coincidence cases can be a bit more tricky but fundamentally proceed in a similar way. We need to show the Fourier transform coefficients of $\chi_1$ and $\chi_2$ both vanish. For collinearity we only need to show that $ \chi_1 \Dot x$ and $ \chi_2 \Dot x $ have the same absolute value.

Pentagon geometry and Jesse Douglas' pentagon construction explained

Let $\omega$ be the simplest primitive $5$th root of unity, with $\ii := \sqrt{-1}$: $$ \omega = \cos\frac {2\pi}{5} + \ii \sin\frac {2\pi}{5}. $$ Note that complex conjugation acts as $\bar{\omega}_5^j = \omega_5^{5-j}$.

The standard pentagons are:

  • a degenerate collection of five points Pentagon chi0
  • the standard regular convex pentagon Pentagon chi1
  • the standard regular pentagram Pentagon chi2
  • the reversed regular pentagram Pentagon chi3
  • the reversed regular convex pentagon Pentagon chi4

These are the 5 standard vectors $$ \chi_0=( 1, 1, 1, 1, 1 ) \quad;\quad \chi_1=( 1, \omega_5^\strut, \omega_5^2, \omega_5^3, \omega_5^4 ) \quad;\quad \chi_2=( 1, \omega_5^2, \omega_5^4, \omega_5^\strut, \omega_5^3 ) \quad;$$

$$ \chi_3=( 1, \omega_5^3, \omega_5^\strut, \omega_5^4, \omega_5^2 ) \quad;\quad \chi_4=( 1, \omega_5^4, \omega_5^3, \omega_5^2, \omega_5^\strut ). $$ which form a basis for $\cplx^5$.

 

Going back to examine the coordinates for $q_i$, the point obtained by extending a line from a vertex to an opposite midpoint by an extra $\frac1{\sqrt5}$, we see $$q_i = -\frac{\sqrt5}5 p_i + \frac{5+\sqrt5}{10} p_{i+2}+ \frac{5+\sqrt5}{10} p_{i+3}.$$ To establish properties, see what components of the various standard types it has. The function $q=(q_0,q_1,q_2,q_3,q_4))\in{\cplx}[{\intg}/5]$ has the harmonic expansion q = \sum_{k=0}^{4} (\chi_k \Dot q) \chi_k = \sum_{k=0}^{4}\hat q_k \chi_k. $$ The character $\chi_0$ is the trivial (or degenerate) character of the trivial representation, and $\hat f_{0}$ is the center of gravity of the values of $f$. The other $\hat f_{1},\hat f_{2},\hat f_{3},\hat f_{4}$ are the harmonic components of $f$.

The center of gravity is just an expression of where the geometric object is located, so we don't expect any special properties for it. But look at $\hat q_2$; we find by calcualtion

$$ \begin{eqnarray} 5\hat {q_2}=5 \chi_2 \Dot q = 0\end{eqnarray} $$

So $\hat q_2$ vanishes identically. Thus there is no regular pentagram component of the polygon $q$. Similarly look at $\hat q_3$, or at

$$ \begin{eqnarray} 5\hat { q_3} = 5\ \chi_3 \Dot q = 5\ \overline{ \overline q\Dot\chi_2} = 0 . \end{eqnarray} $$

We have thus shown that $q = {\hat q}_0 + {\hat q}_1 \chi_1 + {\hat q}_4\chi_4={\hat q}_0 + {\hat q}_1 \chi_1 + {\hat q}_4 \overline{\chi_1}$. This means that $q$ is an affine image of the regular convex pentagon.

Similarly the polygon $r$ constructed is the affine image of a regular pentagram, for like reasons.

Proof of Napoleon's Theorem

Again take $\omega := \omega_3^{\strut} = \frac12(-1 + \ii \sqrt3)$ until further notice, so $\omega^3 - 1 = 0 = 1 + \omega + \omega^2.$ Start by expressing the construction steps in the Napoleon theorem with the help of the characters we introduced. We need to specify an equilateral triangle constructed positively on an oriented segment. The standard unit equilateral triangle is $\chi_{3,1}=:\chi_1$. Its right-hand vertex is at the point $1\in\cplx$; we see that $\chi_1-\chi_0$ is a similar triangle shifted so that its right-hand vertex is at the origin. A diagram may help.

Arithemtic with the cube root of unity.

To shift it so that its first side is aligned positively along the horizontal $x$-axis we multiply the whole by $(\omega-1)^{-1}=3^{-1}(\omega^2-1)$ to get $3^{-1}(\omega^2-1)(\chi_1-\chi_0)$. The triangular building block we need is thus $$ 3^{-1}(\omega^2-1)(\chi_1-\chi_0). $$ There is an important thing to note about the constructed triangle: it is positively oriented with respect to the base interval from $0$ to $1$. It is on the left as we go from $0$ to $1$. To get the negatively oriented block on the right as we go from $0$ to $1$ we need $$ 3^{-1}(\omega-1)(\chi_2-\chi_0). $$

In general, to translate a whole triangular figure by a vector $z$ we have to add $z\chi_0$ to it.

If the segment on which we need to erect a triangle is from $z_j$ to $z_k$, then it is of length $|z_k-z_j|$ and in the direction $\arg(z_k-z_j)$ from the starting point $z_j$. Thus the triangle on the left side of the interval which is to be constructed is $$ 3^{-1}(\omega^2-1)(z_k-z_j)(\chi_1-\chi_0) + z_j\chi_0, $$ and the triangle on the right side is $$ 3^{-1}(\omega-1)(z_k-z_j)(\chi_2-\chi_0) + z_j\chi_0. $$

Finally, to find the centroid of a triangle $z=(z_0,z_1,z_2)$ we just have to take the inner product with $\chi_0$, namely $\chi_0 \Dot z$. So the centroid of the left-side resulting triangle in the last paragraph is, using the fact that the basis vectors $\chi_j$ are orthonormal, \begin{eqnarray*} \chi_0 \Dot( 3^{-1}(\omega^2-1)(z_k-z_j)(\chi_1-\chi_0) + z_j\chi_0) &=& -3^{-1}(\omega^2-1)(z_k-z_j) + z_j\\ &=& -3^{-1}\{(\omega^2-1)z_k-(\omega^2-1)z_j-3z_j\}\\ &=& -3^{-1}\{(\omega^2-1)z_k-(\omega^2+2)z_j\} . \end{eqnarray*}

Theorem 1 (Napoleon) For any nondegenerate triangle in the plane the result of constructing the centroids of equilateral triangles erected externally on the sides of the triangle is three points at the vertices of an equilateral triangle.

Proof. Let the initial triangle be $z=(z_0,z_1,z_2)$, and suppose it positively oriented. Start with externally constructed equilateral triangles. That means we need right-side building blocks. By the above remarks the externally constructed vertices are $$ ( -3^{-1}\{(\omega-1)z_{j+1}-(\omega+2)z_j\} , j\in \{0,1,2\}), $$ so the triangle is $$3 ^{-1}\{(1-\omega)J z+ (\omega+2)z\}. $$ To show that this is an equilateral triangle we take the inner product with $\chi_2$. We know that the original triangle has its Fourier decomposition $z=\hat z_0\chi_0+\hat z_1\chi_1+\hat z_2\chi_2$, and the basis $(\chi_j,j\in\{0,1,2\})$ is orthonormal. Now we see quickly that the $\chi_j$ are eigenvectors of $J$, namely $J\chi_j = \omega^j\chi_J$. Therefore $Jz=\hat z_0\chi_0+\omega \hat z_1\chi_1+\omega^2 \hat z_2\chi_2$. Thus \begin{eqnarray*} \langle\chi_2\Dot3^{-1}\{(1-\omega)J z+ (\omega+2)z\}\rangle &=&3^{-1}\{(1-\omega)\omega^2 \hat z_2+ (\omega+2)\hat z_2\}\\ &=&3^{-1}\{\omega^2-1+\omega +2\}\hat z_2 \\ &=&3^{-1}\{\omega^2+\omega +1\}\hat z_2 \\ &=& 0. \end{eqnarray*} We have shown that constructing the centroids of equilateral triangles externally to the positively oriented initial triangle yields a sequence of points making up the vertices of a positively oriented equilateral triangle.

To deal with left-hand side, i.e. , ‘internal’ equilateral triangles, we just have to do a similar calculation.

Since we may suspect that the resulting vertices will have the reverse orientation we will take the inner product with $\chi_1$. As suspected, they are the vertices of a reverse equilateral triangle. ▓

Corollary The Napoleon construction provides a geometrical way to find the discrete Fourier transform of a triangle.
Proof. First for the external triangles \begin{eqnarray*} \chi_1\Dot(3^{-1}\{(1-\omega)J z+ (\omega+2)z\}) &=&3^{-1}\{(1-\omega)\omega \hat z_1+ (\omega+2)\hat z_1\}\\ &=&3^{-1}\{\omega-\omega^2 +\omega +2\}\hat z_1 \\ &=&3^{-1}\{-\omega^2+2\omega +2\}\hat z_1 \\ &=&3^{-1}\{-3\omega^2\}\hat z_1\\ &=& -\omega^2 \hat z_1, \end{eqnarray*} and \begin{eqnarray*} \langle\chi_0|3^{-1}\{(1-\omega)J z+ (\omega+2)z\}\rangle &=&3^{-1}\{(1-\omega) \hat z_0+ (\omega+2)\hat z_0\}\\ &=&\hat z_0 \end{eqnarray*} (which last we really already knew). Geometrically this means that the external Napoleon triangle has equal sides of lengths $\sqrt3|\hat z_1|$, and if we want to work out the argument of $\hat z_1$ then all we have to do is to rotate the first side of it back by $2\pi/3$ and flip.

From the internal Napoleon triangle we get similarly \begin{eqnarray*} \langle\chi_2|3^{-1}\{(1-\omega^2)J z+ (\omega^2+2)z\}\rangle &=&3^{-1}\{(1-\omega^2)\omega^2 \hat z_2+ (\omega^2+2)\hat z_2\}\\ &=&3^{-1}\{\omega^2-\omega +\omega^2 +2\}\hat z_2 \\ &=& -\omega \hat z_2, \end{eqnarray*} giving us the value for $\hat z_2$. ▓

A generalization of this results from noticing that one could just as well use $N$ as $3$ above for the number of sides, $\chi_{N,k}$ for the initial $N$-gon and some other $\chi_{N,l}$-gon for the constructed figure on each side and still get the same results with careful manipulation of the indices. It is probably helpful to remark that the $N=4$ case says that constructing squares on the sides of a parallelogram gives four centers of gravity which are the vertices of a square, before the general formulation that follows. Let this theorem indicate that one can go a lot further with these methods.

Theorem 2 (Generalized Napoleon) Let $N$ be an integer greater than $2$, and $k$ and $l$ be integers between $1$ and $N-1$. For any nondegenerate affine image of a regular $(N,k)$-gon in the plane, the result of constructing the centroids of regular $(N,l)$-gons externally upon the sides of the initial $(N,k)$-gon is the set of vertices of a regular $(N,k)$-gon if $k^2=l \pmod N$; if the additional figures are constructed internally then a regular $(N,N-k)$-gon results.

Summary

We have seen that there's a relationship between some elementary geometry of the plane and the Discrete Fourier Transform: Assertions of triangle geometry are statements about the harmonics that remain after a prescribed construction, and Napoleon's Theorem gives a geometrical way of constructing a Discrete Fourier Transform of order $3$. This is just a suggestion of some ramifications of this which include relationships to polynomials, circulant matrices, interpolation and splines, Siebeck's and Marden's theorems on polynomial zeroes, and the Heisenberg-Weyl group as it occurs in signal processing.

Acknowledgements

I would like to thank Bill Casselman and David Austin for many improvements to the exposition, Davide Cervone and Robert Miner for the excellent tool MathJax, and Markus Hohenwarter and his team for the valuable Geogebra package. Mathematical Reviews and the University of Michigan have provided unparalleled access to the literature of mathematics.

References

  • Douglas, Jesse
    1. Geometry of polygons in the complex plane. J. Math. Phys. Mass. Inst. Tech. 19 (1940), 93.130. MR 0001574 (1,261f)
    2. On linear polygon transformations. Bull. Amer. Math. Soc. 46 (1940), 551.560. MR 0002178 (2,9i)
    3. A theorem on skew pentagons. Scripta Math. 25 1960 5.9. MR 0117643 (22 #8419)
    The early papers on the whole approach to geometry through the Discrete Fourier Transform are items 1 and 2. The third introduces the pentagon construction discussed above, and is dedicated to a long-time colleague geometer, Jekuthiel Ginsburg. Douglas was awarded one of the first two Fields medals in 1936.
  • Schoenberg, I. J.
    1. The finite Fourier series and elementary geometry. Amer. Math. Monthly 57 (1950), 390.404. MR 0036332 (12,92f)
    2. Mathematical time exposures. Mathematical Association of America, Washington, DC, 1982. ix+270 pp. ISBN: 0-88385-438-4. MR 0711022 (85b:00001)
    3. The harmonic analysis of skew polygons as a source of outdoor sculptures. The geometric vein, pp. 165.176, Springer, New York-Berlin, 1981. MR 0661776 (84d:52011)
    Article 1 discusses these matters very clearly, as do several chapters in the book 2. Schoenberg actually did make a sculpture based on Douglas's construction described in 3.
  • Andreescu, Titu; Andrica, Dorin, Complex numbers from A to... Z. Translated and revised from the 2001 Romanian original. Birkh.user Boston, Inc., Boston, MA, 2006. xiv+321 pp. ISBN: 978-0-8176-4326-3; 0-8176-4326-5. MR 2168182
    An introductory text using complex coordinate methods with an eye toward mathematical Olympiad problems.
  • Web sites:
    Cut the Knot: Interactive Mathematics Miscellany and Puzzles, http://www.cut-the-knot.org/
    An excellent source for interactive geometry and, in particular, proofs of Morley's theorem.
  • http://www-personal.umich.edu/~pion/WebGeom/
    The author's personal site where more related material is to be posted.

Patrick D. F. Ion
Mathematical Reviews
ion at ams.org email icon

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