The Maths of Pylons,
Art Galleries and Prisons Under
the Spotlight
John D. Barrow
Some Fascinating Properties
of Straight Lines
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Draw 4 lines through
all 9 points
The pencil must not
leave the paper.
No reversing
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Think Outside The Box
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Think Outside The Box
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Think Outside The Box
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Think Outside The Box
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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!
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.
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
Orthogonal galleries
All corners are right angles
Only [n/4] guards are needed, and are always sufficient
n = 100 needs only 25 guards now
Rectangular galleries
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
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?
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.
Mobile Guards
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
A Worst Case
[(3n+4+4h)/16]
mobile guards
required
n = 20 h = 4 needs
80/16 = 5 guards
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
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 = 4 example
4/2 = 2
x is smallest integer  x
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
Prisoner Cell H Block Problem
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
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