On the Optimality of Napoleon Triangles

On the Optimality of Napoleon Triangles

Abstract

An elementary geometric construction known as Napoleon’s theorem produces an equilateral triangle built on the sides of any initial triangle: the centroids of each equilateral triangle meeting the original sides, all outward or all inward, comprise the vertices of the new equilateral triangle. In this note we observe that two Napoleon iterations yield triangles with useful optimality properties. Two inner transformations result in a (degenerate) triangle whose vertices coincide at the original centroid. Two outer transformations yield an equilateral triangle whose vertices are closest to the original in the sense of minimizing the sum of the three squared distances.

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1 Introduction.

In elementary geometry, one way of constructing an equilateral triangle from any given triangle is as follows: in a plane the centroids of equilateral triangles erected, either all externally or all internally, on the sides of the given triangle form an equilateral triangle, illustrated in Figure 1 [1]. This result is generally referred to as Napoleon’s theorem, notwithstanding its dubious origins — see [2] for a detailed history of the theorem. We will refer to these constructions as the outer and inner Napoleon transformations and the associated equilateral triangles as the outer and inner Napoleon triangles of the original triangle, respectively. Conversely, given its outer and inner Napoleon triangles in position (i.e. they are oppositely oriented and have the same centroid), the original triangle is uniquely determined [3]. A fascinating application of Napoleon triangles is the planar tessellation used by Escher: a plane can be tiled using congruent copies of the hexagon defined by the vertices of any triangle and its outer Napoleon triangle, known as Escher’s theorem [4].

Figure 1: An illustration of (left) the Fermat point , outer Torricelli configuration and outer Napoleon triangle , and (right) inner Torricelli configuration and inner Napoleon triangle of a triangle . Note that centroids of Torricelli configurations, Napoleon triangles and the original triangle all coincide, i.e. .

Equilaterals built on the sides of a triangle make a variety of appearances in the classical literature. Torricelli uses this construction to locate Fermat’s point minimizing the sum of distances to the vertices of a given triangle, now called the Fermat-Torricelli problem [5]. The unique solution of this problem is known as the Fermat-Torricelli point of the given triangle, located as follows [6]. If an internal angle of the triangle is greater than , then the Fermat point is at that obtuse vertex. Otherwise, the three lines joining opposite vertices of the original triangle and externally erected triangles are concurrent, and they intersect at the Fermat point, see Figure 1. The triangle defined by the new vertices of the erected equilateral triangles is referred to as the Torricelli configuration [7, 8].

In this paper we demonstrate some remarkable, but not immediately obvious, optimality properties of twice iterated Napoleon triangles. First, two composed inner Napoleon transformations of a triangle collapse the original to a point located at its centroid which, by definition, minimizes the sum of squared distances to the vertices of the given triangle. Surprisingly, two composed outer Napoleon transformations yield an equilateral triangle optimally aligned with the original by virtue of minimizing the sum of squared distances between the paired vertices. More precisely, for any triangle with the vertices 1 , we will say that the triangle is an optimally aligned equilateral triangle of if it solves the following constrained optimization problem:

(1)

where and denotes the standard Euclidean norm on . As we show below, this optimization problem has a unique solution so long as , , are not collinear.

2 Torricelli and Napoleon Transformations.

For any ordered triple , let denote the rotation matrix corresponding to a counter-clockwise rotation by in the plane, defined by orthonormal vectors and , in which the triangle formed by is positively oriented (i.e. its vertices in counter-clockwise order follow the sequence ), 2

(2)

where 3

(3)

where is the projection onto (the tangent space of at point ), and is the identity matrix. Note that is both positively and negatively oriented if is collinear. Consequently, to define a plane containing such we select an arbitrary vector perpendicular to in (3). It is also convenient to have denote the centroid of , i.e. .

In general, the Torricelli and Napoleon transformations of three points in Euclidean -space can be defined based on their original planar definitions in a 2-dimensional subspace of containing . That is to say, for any , select a 2-dimensional subspace of containing , and then construct the erected triangles on the side of in this subspace to obtain the Torricelli and Napoleon transformations of . Accordingly, let and denote the Torricelli and Napoleon transformations where the sign, and , determines the type of the transformation, inner and outer, respectively. One can write closed-form expressions of the Torricelli and Napoleon transformations as:

Lemma 1.

The Torricelli and Napoleon transformations of any triple on a plane containing are, respectively, given by 4

(4)
(5)

where

(6)
Proof.

One can locate the new vertex of an equilateral triangle, inwardly or outwardly, constructed on one side of in the plane containing using different geometric properties of equilateral triangles. We find it convenient to use the perpendicular bisector of the corresponding side of , the line passing through its midpoint and being perpendicular to it, such that the new vertex is on this bisector and a proper distance away from the side of .

For instance, let . Consider the side of joining and , using the bisector, , to locate the new vertex, , of inwardly erected triangle on this side as

(7)

where (2) defines a counter-clockwise rotation by in the plane where is positively oriented. Note that the height of an equilateral triangle from any side is times its side length. Hence, by symmetry, one can conclude (4).

Given a Torricelli configuration , by definition, the vertices of associated Napoleon triangle are given by

(8)

which is equal to (5), and so the result follows. ∎

Note that the Torricelli and Napoleon transformations of are unique if and only if is non-collinear. If, contrarily, is collinear, then is both positively and negatively oriented and for there is more than one 2-dimensional subspace of containing .

Remark 1 ([3]).

For any , the centroid of the Torricelli configuration , the Napoleon configuration and the original triple all coincide,

(9)

and distances between the associated elements of and are all the same, i.e. for any

(10)

An observation key to all further results is that Napoleon transformations of equilateral triangles are very simple.

Lemma 2.

The inner Napoleon transformation of any triple comprising the vertices of an equilateral triangle collapses it to the trivial triangle all of whose vertices are located at its centroid ,

(11)

whereas the outer Napoleon transformation reflects the vertices of with respect to its centroid , 5

(12)
Proof.

Observe that the inwardly erected triangle on any side of an equilateral triangle is equal to the equilateral triangle itself, i.e. , and so, by definition, one has (11). Alternatively, using (5), one can obtain

(13)

where is defined as in (6).

Now consider outwardly erected equilateral triangles on the sides of an equilateral triangle, and let . Note that each erected triangle has a common side with the original triangle. Since is equilateral, observe that the midpoint of the unshared vertices of an erected triangle and the original triangle is equal to the midpoint of their common sides, i.e. and so on. Hence, we have . Thus, one can verify the result using (5) as

(14)

Since the Napoleon transformation of any triangle results in an equilateral triangle, motivated from Lemma 2, we now consider the iterations of the Napoleon transformation. For any let denote the -th Napoleon transformation defined to be

(15)

where we set , and is the identity map on .

It is evident from Lemma 2 that:

Lemma 3.

For any and ,

(16)

As a result, the basis of iterations of the Napoleon transformations consists of and , whose explicit forms, except , are given above. Using (5) and (12), the closed-form expression of the double outer Napolean transformation can be obtained as:

Lemma 4.

An arbitrary triple, gives rise to the double outer Napoleon triangle, , according to the formula

(17)
Proof.

By Napoleon’s theorem, is an equilateral triangle. Using (5) and Lemma 2, one can obtain the result as follows:

(18)
(19)

where is defined as in (6). ∎

Notice that is a convex combination of and , see Figure 2.


Figure 2: (left) Outer, , and double outer, , Napoleon transformations of a triangle . (right) The double outer Napoleon triangle is a convex combination of the original triangle and its inner Torricelli configuration .

3 Optimality of Napoleon Transformations.

To best of our knowledge, the Napoleon transformation is mostly recognized as being a function into the space of equilateral triangles. In addition to this inherited property, has an optimality property that is not immediately obvious. Although the double inner Napoleon transformation is not really that interesting to work with, it gives a hint about the optimality of : for any given triangle yields a trivial triangle all of whose vertices are located at the centroid of the given triangle which, by definition, minimizes the sum of squared distances to the vertices of the original triangle. Surprisingly, one has a similar optimality property for :

Theorem 1.

The double outer Napoleon transformation (17) yields the equilateral triangle most closely aligned with in the sense that it minimizes the total sum of squared distances between corresponding vertices. That is to say, for any , is an optimal solution of the following problem

(20)

where . Further, if is non-collinear, then (20) has a unique solution.

Proof.

Using the method of Lagrange multipliers [10], we first show that an optimal solution of (20) lies in the plane containing the triangle . Then, to show the result, we solve (20) using a proper parametrization of equilateral triangles in .

The Lagrangian formulation of (20) minimizes

(21)

where are Lagrange multipliers. Necessary conditions for the locally optimal solutions of (20) is 6

(22)

from which one can conclude that an optimal solution of (20) lies in the plane containing . Accordingly, without loss of generality, suppose that is a positively oriented triangle in , i.e. its vertices are in counter-clockwise order in .

In general, an equilateral triangle in with vertices can be uniquely parametrized using two of its vertices, say and , and a binary variable specifying the orientation of ; for instance, if is positively oriented, and so on. Consequently, the remaining vertex, , can be located as

(23)

where is the rotation matrix defining a rotation by .

Hence, one can rewrite the optimization problem (20) in term of new parameters as an unconstrained optimization problem as: for and ,

(24)

where , and is the identity matrix. Note that , and .

For a fixed , (24) is a convex optimization problem of and because every norm on is convex and compositions of convex functions with affine transformations preserve convexity [11]. Hence, a global optimal solution of (24) occurs where the gradient of the objective function is zero at,

(25)

which simplifies to

(26)

Note that the objective function, , is strongly convex since its Hessian, , satisfies

(27)

which means that for a fixed the optimal solution of (24) is unique.

Now observe that

(28)

hence the solution of linear equation (26) is

(29)
(30)

Here, substituting and back into (23) yields

(31)

Thus, overall, we have

(32)

where and are defined as in (6). Recall that is assumed to be positively oriented, i.e. , and so it is convenient to have the results in terms of Torricelli transformations (4). As a result, the difference of and is simply given by

(33)

Finally, one can easily verify that the optimum value of is equal to since the distance of to its inner Torricelli configuration is always less than or equal to its distance to the outer Torricelli configuration . Here, the equality only holds if is collinear. Thus, an optimal solution of (20) coincides with the double outer Napoleon transformation, (17), and it is the unique solution of (20) if is non-collinear. ∎

As a final remark we would like to note that our particular interest in the optimality of Napoleon triangles comes from our research on coordinated robot navigation, where a group of robots require to interchange their (structural) adjacencies through a minimum cost configuration determined by the double outer Napoleon transformation [12].

{acknowledgment}

Acknowledgment. We would like to thank Dan P. Guralnik for the numerous discussions and kind feedback. This work was funded in part by the Air Force Office of Science Research under the MURI FA9550-10-1-0567.

Footnotes

  1. Here, denotes the set of real numbers, and is the -dimensional Euclidean space.
  2. denotes the transpose of matrix .
  3. For any trivial triangle all of whose vertices are located at the same point we fix by setting whenever .
  4. Here, denotes the Kronecker product [9].
  5. Here, is the column vector of all ones, and denotes the standard array product.
  6. Here, denotes the gradient taken with respect to the coordinate .

References

  1. H. S. M. Coxeter and S. L. Greitzer, Geometry revisited, vol. 19. Mathematical Association of America, 1996.
  2. B. Grünbaum, “Is Napoleon’s theorem really Napoleon’s theorem?,” The American Mathematical Monthly, vol. 119, no. 6, pp. 495–501, 2012.
  3. J. E. Wetzel, “Converses of Napoleon’s theorem,” The American Mathematical Monthly, vol. 99, no. 4, pp. 339–351, 1992.
  4. J. Rigby, “Napoleon, Escher, and tessellations,” Mathematics Magazine, vol. 64, no. 4, pp. 242–246, 1991.
  5. Y. S. Kupitz, H. Martini, and M. Spirova, “The Fermat–Torricelli problem, part i: A discrete gradient-method approach,” Journal of Optimization Theory and Applications, pp. 1–23, 2013.
  6. S. Gueron and R. Tessler, “The Fermat–Steiner problem,” The American Mathematical Monthly, vol. 109, no. 5, pp. 443–451, 2002.
  7. H. Martini and B. Weissbach, “Napoleon’s theorem with weights in n-space,” Geometriae Dedicata, vol. 74, no. 2, pp. 213–223, 1999.
  8. M. Hajja, H. Martini, and M. Spirova, “New extensions of Napoleon’s theorem to higher dimensions,” Beitr. Algebra Geom, vol. 49, no. 1, pp. 253–264, 2008.
  9. A. Graham, Kronecker products and matrix calculus: with applications, vol. 108. Horwood Chichester, 1981.
  10. D. P. Bertsekas, Nonlinear Programming. Athena Scientific, 1999.
  11. S. P. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.
  12. O. Arslan, D. Guralnik, and D. E. Koditschek, “Navigation of distinct euclidean particles via hierarchical clustering,” in Algorithmic Foundations of Robotics XI, vol. 107 of Springer Tracts in Advanced Robotics, pp. 19–36, 2015.
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