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 form 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.

Part I. The theorems

Theorems of Coincidence: Triangle Medians

The median lines of a triangle are the lines drawn from each vertex to the mid-point 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).

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.

Equilateralness - Napoleon's Theorem

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

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.

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.

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.

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.

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.

If we separate the two triangles by making the standard one larger, say lying on a circle of radius $\lambda>1$ then we get

which shows the triangle is an isosceles one with height $\sfrac34(\lambda+1)$ lying along the real axis.

If we separate the two basic triangles $\chi_1$ and $\chi_2$ by turning them in opposite directions by a turn $\ee^{\ii \alpha}$ (in the attractive terminology of Frank Morley for complex numbers of modulus 1) to make $$ \begin{eqnarray} {\textstyle\sfrac12}(\ee^{\ii \alpha}\chi_1+\ee^{-\ii \alpha}\chi_2)&=& {\textstyle\sfrac12}\left\{\ee^{\ii \alpha}(1,-{\textstyle\sfrac12}+{\textstyle\sfrac{\sqrt3}2}\ii, -{\textstyle\sfrac12}-{\textstyle\sfrac{\sqrt3}2}\ii) +\ee^{-\ii \alpha}(1,-{\textstyle\sfrac12}-{\textstyle\sfrac{\sqrt3}2}\ii, -{\textstyle\sfrac12}+{\textstyle\sfrac{\sqrt3}2}\ii)\right\}\cr &=& \left({\textstyle\sfrac12}(\ee^{\ii \alpha}+\ee^{-\ii \alpha}), -{\textstyle\sfrac14}(\ee^{\ii \alpha}+\ee^{-\ii \alpha})+ {\textstyle\sfrac{\sqrt3}4}\ii(\ee^{\ii \alpha}-\ee^{-\ii \alpha}) ,-{\textstyle\sfrac14}(\ee^{\ii \alpha}+\ee^{-\ii \alpha})- {\textstyle\sfrac{\sqrt3}4}\ii(\ee^{\ii \alpha}-\ee^{-\ii \alpha})\right)\cr &=&(\cos \alpha, -{\textstyle\sfrac12}\cos \alpha-{\textstyle\sfrac{\sqrt3}2}\sin \alpha, -{\textstyle\sfrac12}\cos \alpha+{\textstyle\sfrac{\sqrt3}2}\sin \alpha) ) \end{eqnarray} $$ $$ \begin{eqnarray} {\textstyle\sfrac12}(\ee^{\ii \alpha}\chi_1+\ee^{-\ii \alpha}\chi_2)= (\cos \alpha, -{\textstyle\sfrac12}\cos \alpha-{\textstyle\sfrac{\sqrt3}2}\sin \alpha, -{\textstyle\sfrac12}\cos \alpha+{\textstyle\sfrac{\sqrt3}2}\sin \alpha) ) \end{eqnarray} $$ then we have achieved a splitting of the double point at $-\sfrac12$ in $\sfrac12(\chi_1+\chi_2)$ but still have a degenerate triangle on the real axis, with the second vertex at $z_1=-\sfrac12\cos \alpha-\sfrac{\sqrt3}2\sin \alpha$ and the third symmetrically placed about $-\frac12\cos\alpha$ at $z_2=-\sfrac12\cos \alpha+\sfrac{\sqrt3}2\sin \alpha$.

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.

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 $\bar{\omega}_5^j = \omega_5^{5-j}$.

These are the 5 standard vectors $$ \chi_0=( 1, 1, 1, 1, 1 ) \quad;\quad \chi_1=( 1, \omega_5^\strut, \omega_5^2, \omega_5^3, \omega_5^4 ) \quad;\quad \chi_2=( 1, \omega_5^2, \omega_5^4, \omega_5^\strut, \omega_5^3 ) \quad;\quad \chi_3=( 1, \omega_5^3, \omega_5^\strut, \omega_5^4, \omega_5^2 ) \quad;\quad \chi_4=( 1, \omega_5^4, \omega_5^3, \omega_5^2, \omega_5^\strut ). $$ which form a basis for $\cplx^5$.

Going back to examine the coordinates for $q_i$, the point got by extending a line from a vertex to an opposite mid-point 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

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

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.

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 $3^{-1}(\omega^2-1)(\chi_1-\chi_0)$. This is clear because we want to move the vertex after the base at $0$, namely at $\omega-1$, to the point $1$, and $$ (\omega -1)(\omega^2-1)=\omega^3-\omega^2-\omega+1=3. $$ Alternatively, we see immediately that $\overline{\omega-1}=\omega^2-1$ and $|\omega-1|=3$. 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. To rotate by a angle $\alpha$ we have to multiply by the scalar turn $\ee^{\ii\alpha}$, and to scale by a homothety $\lambda > 0$ we have to multiply by the scalar $\lambda$. For instance, if we had not multiplied by $(\omega-1)^{-1}$ just a moment ago, but only used the rotation effected by the unit vector $\{3^{-1/2}(\omega-1)\}^{-1}=3^{-1/2}(\omega^2-1)$ then we would have had to scale the building-block triangle to unit side-length by multiplying with $3^{-1/2}$.

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.

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.

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