activity 1-4 : the least road problem
TRANSCRIPT
Activity 1-4 : The Least Road Problem
www.carom-maths.co.uk
Task: you are given four towns at the corner of a square, side 1 km.
It is decided to build a system of roads that will enable any town to be reached from any other.
What is the least amount of road required?
4km
3km
3km
2.828km
Can we do any better than this?
What if we combine the two solutions on the right-hand side?
?
It seems likely that a best solution will be symmetrical.
Autograph File http://www.s253053503.websitehome.co.uk/
carom/carom-files/carom-1-4-1.agg
So for this design, the minimum value for the road length
is 2.732 km.
This is the best we can do!
This happens when θ (see previous page) is exactly 120o.
It turns out that the angle 120o is a common theme
in this type of problem...
What if you have six towns at the corners of a regular hexagon:which pattern would you choose here?
What if you have five towns at the corners of a regular pentagon:which pattern would you choose here?
Suppose you are given three towns at the corners of some
triangle: what is your best strategy here?
If the triangle has its largest angle larger than 120o then simply pick
the two shorter sides to connect the three towns.
If the triangle ABC has its largest angle
smaller than 120o then construct
an equilateral triangle on one side,
and find the 120o point you need as follows...
(This point is called the Steiner Point).
Jakob Steiner,Swiss
(1796 – 1863)
Steiner disliked algebra and analysis and believed that calculation
replaces thinking while geometry stimulates thinking.
(MacTutor).
How about six towns that are at the corners of two squares placed side by side?
We can confirm some of these answers by looking at soap-bubbles.
http://www.youtube.com/
watch?v=dAyDi1aa40E
Numberphile video link
With thanks to:Singing Banana (James Grime)
MacTutor (History of Maths site).
Carom is written by Jonny Griffiths, [email protected]