How many suns need to be placed around a spherical planet so that an observer, standing anywhere on the planet with a clear view of the sky, is guaranteed to see at least one of the suns entirely above the horizon? The answer is four, and this number is called the illumination number of the shape of the planet.
The Illumination Conjecture says that any n-dimensional convex body can be illuminated by at most 2 to the n light sources, and that the maximum number of lights is only needed if the body is a parallelepiped (i.e., an n-dimensional parallelogram).
In geometry, a shape is called convex if the line segment connecting any two points in the shape lies entirely within the shape. In the diagram above, the shape on the left satisfies this definition. However, the shape on the right is not convex, because there exist two points in the set with the property that the line segment connecting them does not lie entirely within the set.
Remarkably, the Illumination Conjecture is related to the problem of covering a shape with smaller copies of itself. As illustrated below, a triangle can be covered by three smaller copies of itself, but a square needs four smaller copies. It is a theorem in geometry that any two-dimensional convex shape can be covered by at most four smaller copies of itself, and furthermore, that four copies are only necessary if the shape is a parallelogram.
The Hadwiger Conjecture in geometry is the n-dimensional analogue of this theorem. The conjecture states that any convex shape in n-dimensional Euclidean space can be covered by at most 2 to the n smaller copies of itself, and furthermore, that this maximum number is only needed if the shape is a parallelepiped. The 2-dimensional case was proved by F.W. Levi in 1955, but even now, the 3-dimensional case has not been proved in full generality.
Let C(K) be the number of smaller copies of K that are necessary to cover K, so that for example C(K)=4 if K is a square, and C(K)=3 if K is a triangle. It is a theorem, even in n dimensions, that C(K) is also equal to the number of copies of the interior of K that are necessary to cover K. Here, “interior” refers to the inside of the shape, i.e., the shape with its boundary removed. More surprisingly, the number C(K) is also equal to the illumination number of K. The Hadwiger Conjecture and the Illumination Conjecture are therefore equivalent, in the sense that they are either both true, or both false.
So why does it take four lights to illuminate a sphere completely? Clearly, one light source will not be enough, since half the sphere would be in darkness, as in the picture above. A similar problem occurs with two light sources that are close together, like in the Star Wars picture. Even if the two light sources are as far apart as possible, for example above the north and south poles of the planet, two lights are not sufficient, because an observer standing on the equator will not see either light source strictly above the horizon. For three light sources, one could try placing the lights above the equator at 120 degree intervals, but still, to an observer standing at the north or south pole, all of the lights would be exactly on the horizon. With four light sources, things can be made to work, for example by putting the four lights at the corners of a tetrahedron, and putting the sphere in the centre of the tetrahedron, as in the arrangement below.
More generally, it is a theorem that if K is an n-dimensional ball, or in fact any smooth convex n-dimensional body, then K can be illuminated by n+1 lights. The scenario with the spherical planet is a special case of this, with n=3. In 1984, Marek Lassak proved that the Illumination Conjecture is true in the case where K is a centrally symmetric body, meaning that if a point lies in K, then so does the point diametrically opposite it.
I found out about this problem from the recent paper On the multiple illumination numbers of convex bodies by Kirati Sriamorn. The author considers a more general version of the problem, in which each observer needs to be able to see at least m light sources from any point on the surface. The paper proves that for a 2-dimensional smooth convex shape K, at least 2m+1 lights are necessary to illuminate the boundary of K in this way. The author conjectures that for a general n-dimensional convex body, at most 2ⁿm (m times 2 to the n) light sources are needed, and that the maximum number is only needed for a parallelepiped.
Picture credits and relevant links
The Hadwiger conjecture is named after Hugo Hadwiger (1908-1981), which he posed as a problem in 1957. Hadwiger’s name is associated with some other open problems in mathematics, including Hadwiger’s conjecture in graph theory, which is a generalization of the Four Colour Theorem.
The still of the iconic Binary Sunset scene from Star Wars: Episode IV – A New Hope was enhanced by user NonPrime on Reddit (/r/StarWars).
The picture of the triangles and squares is by David Eppstein and appears on Wikipedia’s page on the Hadwiger Conjecture.
The pictures of the convex and non-convex sets are by CheCheDaWaff and Oleg Alexandrov and appear on Wikipedia’s page on convex sets.
The picture of the sphere with the shadow is by Martin Kraus.
The tetrahedron picture is by Benjah-bmm27.
Really interesting! I will never view science fiction movies with planets that have multiple suns the same way ever again. 👍👍