Post on 17-Dec-2015
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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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
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 …
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
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
4
2 3
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Prisoner Cell H Block 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