The Maths of The Maths of PylonsPylons,,

Art Galleries and Art Galleries and Prisons Under the Prisons Under the

SpotlightSpotlightJohn D. BarrowJohn D. Barrow

Some Fascinating Properties Some Fascinating Properties of Straight Linesof Straight Lines

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Draw 4 lines throughall 9 pointsThe pencil must notleave the paper.No reversing

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Pylon of the Month

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

RigidityRigidity

The triangle is the The triangle is the ONLY ONLY

rigid polygonrigid polygon

ALLALL convex polyhedra are convex polyhedra are rigidrigid

What about non-convex What about non-convex polyhedra?polyhedra?

Robert Connelly (1978) finds anRobert Connelly (1978) finds an18 triangular-sided example18 triangular-sided example

That keeps the volume the sameThat keeps the volume the sameBUTBUT

““Almost every” non-convex Almost every” non-convex polyhedron is rigidpolyhedron is rigid

Klaus Steffen’s 14-sided rigid Klaus Steffen’s 14-sided rigid non-convex polyhedron non-convex polyhedron

with 9 vertices and 21 edgeswith 9 vertices and 21 edges

Guarding Art GalleriesGuarding Art Galleries

The Art Gallery ProblemThe Art Gallery Problem

camera

How many cameras are needed to guard a gallery and where should they be placed?

Simple Polygonal GalleriesSimple 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 ?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!

TriangulationTriangulationTo make things easier, we divide a polygon into pieces that eachneed one guard

Join pairs of corners by non-intersectinglines that lie inside the polygon

Guard the galleryby placing a camera in every triangle

3-Colouring the Gallery3-Colouring the Gallery

Assign each corner a colour: pink, green, or yellow.

Any two corners connected by an edge or a diagonal must havedifferent colours. n = 19Thus the vertices of every trianglewill be in three different colors.

If a 3-colouring is possible, put guards at corners of same colourPick 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 The Chvátal Art Gallery TheoremTheorem

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 = 33.33 and [n/3] = 33[x] is the integer part of x

The Worst Case ScenarioThe 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 roomsand 4 guardsin 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 ProblemHow 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 wallsDiagonal 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 = 0needs [304/16] = 9, whereas with immobile guards it is [n/4] =25

Mobile GuardsCounter egCounter eg

[(3n+4+4h)/16] mobile guards

required

n = 20 h = 4 needs80/16 = 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 ProblemThe Fortress Problem

n/2n/2 corner guards are always necessary and corner guards are always necessary and sufficient to guard the exterior of a polygonal sufficient to guard the exterior of a polygonal

fortress with n wallsfortress with n walls

n = 4 example4/24/2 = 2 = 2

xx is smallest integer is smallest integer x xSo So = 4 and = 4 and 22 = = 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 nCorners: 1 + n/4 corner guards are necessary and sufficient

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

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Prisoner Cell H Block Problem

The Prison Yard ProblemThe Prison Yard Problem

The Prison Yard ProblemThe Prison Yard Problem

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

It is now time for you to return to your cells