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The Maths of Pylons , Art Galleries and Prisons Under the Spotlight

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### What about non-convex polyhedra?

O O O

O O O

O O O

O O O

O O O

Draw 4 lines through

all 9 points

The pencil must not

leave the paper.

No reversing

O O O

O O O

O O O

O O O

O O O

O O O

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

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

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

Guard the gallery

by placing a camera in

every triangle

Join pairs of corners by non-intersecting

lines that lie inside the polygon

3-Colouring the Gallery

Assign each corner a colour:

pink,green,oryellow.

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.

For n = 100, n/3 = 33.33 and [n/3] = 33

[x] is the integer part of x

Note that [n/3] cameras may not always be necessary

Finding the minimum number is computationally ‘hard’.

The Worst Case Scenario

[n/3] V-shaped rooms

A camera can never be positioned

so as to watch over two Vs

Here, the maximum of [n/3] cameras is required

All corners are right angles

Only [n/4] guards are needed, and are always sufficient

n = 100 needs only 25 guards now

All adjacent rooms have connecting archways

8 rooms

and 4 guards

in the arches

In a rectangular gallery with r rooms,

[r/2] guards are needed to guard the gallery

How many guards must be placed in the gallery so that at least m guards

are visible from every point in the gallery?

m = 2: A polygon which requires 4 guards to provide double coverage.

The entire polygon is only visible from the vertex

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.

Counter eg

Counter eg

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

An orthogonal gallery divided into rectangular rooms

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

Find the shortest closed route around

the gallery such that every point can

be seen at least once

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 = 4 example

4/2 = 2

x is smallest integer x

So = 4 and 2 = 2

n/2 corner guards orn/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

4

1

For orthogonal fortresses with n

Corners: 1 + n/4 corner guards are

necessary and sufficient

2

3

12-sided H block will need 1 + 3 = 4

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

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