The Maths of Pylons, Art Galleries and Prisons Under the Spotlight John D. Barrow

Some Fascinating Properties of Straight Lines O OO

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Pylon of the Month

“PGG 3 and friends gather around National Grid Company Plc's Norwich Main Substation”

Rigidity 

The triangle is the ONLY rigid polygon

ALL convex polyhedra are rigid

What about non-convex polyhedra? Robert Connelly (1978) finds an 18 triangular-sided example That keeps the volume the same BUT “Almost every” non-convex polyhedron is rigid

Klaus Steffen’s 14-sided rigid non-convex polyhedron with 9 vertices and 21 edges

Guarding Art Galleries

The Art Gallery Problem camera How many cameras are needed to guard a gallery and where should they be placed?

Simple Polygonal Galleries Regions with holes are not allowed and no self intersections convex polygon one camera is enough an arbitrary n-gon (n corners) ? cameras might be needed

How Many Cameras ? n – 2 cameras can guard the simple n-sided polygon. A camera on a diagonal guards two triangles.  no. cameras can be reduced to roughly n/2. A corner is adjacent to many triangles. So placing cameras at vertices can do even better …

Triangulate!

Triangulation To make things easier, we divide a polygon into pieces that each need one guard Join pairs of corners by non-intersecting lines that lie inside the polygon Guard the gallery by placing a camera in every triangle

3-Colouring the Gallery Assign each corner a colour: pink, green, or yellow. Any two corners connected by an edge or a diagonal must have different colours. n = 19 Thus the vertices of every triangle will be in three different colors. If a 3-colouring is possible, put guards at corners of same colour Pick the smallest of the coloured corner groupings to locate the cameras. You will need at most [n/3] = 6 cameras where [x] is the integer part of x.

The Chvátal Art Gallery Theorem For a simple polygon with n corners, [n/3] cameras are sufficient and sometimes necessary to have every interior point visible from at least one of the cameras. Note that [n/3] cameras may not always be necessary Finding the minimum number is computationally ‘hard’. For n = 100, n/3 = and [n/3] = 33 [x] is the integer part of x

The Worst Case Scenario [n/3] V-shaped rooms Here, the maximum of [n/3] cameras is required A camera can never be positioned so as to watch over two Vs

All corners are right angles Only [n/4] guards are needed, and are always sufficient n = 100 needs only 25 guards now Orthogonal galleries

In a rectangular gallery with r rooms, [r/2] guards are needed to guard the gallery Rectangular galleries All adjacent rooms have connecting archways 8 rooms and 4 guards in the arches

We can find a gallery which can be covered by one guard located at a particular point, but if the guard is placed elsewhere, even arbitrarily close to the first guard, some of the gallery will be hidden when the guard is at the new position. m = 2: A polygon which requires 4 guards to provide double coverage. The entire polygon is only visible from the vertex The Double Cover Problem How many guards must be placed in the gallery so that at least m guards are visible from every point in the gallery?

Edge guards patrol along the polygon walls Diagonal guards patrol inside the gallery along straight lines between corners In 1981, Toussaint conjectured that except for a small number of polygons, [n/4] edge guards are sufficient to guard a polygon. Still unproven. O’Rourke proved that the minimum number of mobile guards necessary and sufficient to guard a polygon is [n/4]. He also showed that [(3n+4+4h)/16] mobile guards are necessary and sufficient to guard orthogonal polygons with h holes. n = 100 and h = 0 needs [304/16] = 9, whereas with immobile guards it is [n/4] =25 Mobile Guards Counter eg

[(3n+4+4h)/16] mobile guards required n = 20 h = 4 needs 80/16 = 5 guards A Worst Case

More than [n/2] guards may be needed. Take a central rectangular room with a similar room on each side. One guard can watch the central room and one other. But no two side rooms share a common wall so each need an extra guard. So, five rooms require four guards. For a gallery with c corners and h holes that is divided into r rectangular rooms, we may need [(2r +c - 2h - 4) / 4] guards Here: c = 20, h = 4, r = 5 so [18/4] = [9/2] = 4 An orthogonal gallery divided into rectangular rooms

The Night Watchman’s Problem Find the shortest closed route around the gallery such that every point can be seen at least once

The Art Thief’s Problem Find the shortest path around the gallery that is not visible from particular security points

The Fortress Problem  n/2  corner guards are always necessary and sufficient to guard the exterior of a polygonal fortress with n walls  n/2  corner guards are always necessary and sufficient to guard the exterior of a polygonal fortress with n walls n = 4 example  4/2  = 2  x  is smallest integer  x So  = 4 and  2  = So  = 4 and  2  = 2

 n/2  corner guards or  n/3  point guards (ie located anywhere) are always sufficient and sometimes necessary to guard the polygonal exterior of a fortress with n corners n = 7 needs  7/3  = 3 point guards and 4 corner guards

For orthogonal fortresses with n Corners: 1 +  n/4  corner guards are necessary and sufficient 12-sided H block will need = Prisoner Cell H Block Problem

The Prison Yard Problem

Suppose you want to guard the interior and the exterior  n/2  corner guards are always sufficient and may be necessary for a convex polygon with n corners. It is [n/2] if non convex. Eg n = 101: 51 for convex and 50 for non convex [5n/12] + 2 corner guards or [(n+4)/3] point guards are always sufficient for an orthogonal prison with holes Eg n = 100: 43 corner guards or 34 point guards suffices

It is now time for you to return to your cells

All the convex polyhedra with regular polygonal faces

Illumination by mirrors Unknown if all polygons with mirror surfaces can be lit from a single source. It is untrue for general shapes from the centre pt but not from another pt here are 2 semi-circles with common centre + 2 more

3d polyhedral gallery where a guard at each corner cannot see every point in the interior