activity 1-4 : the least road problem

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, hello@jonny-griffiths.net