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Number Theory/Tunnel Shapes.


Dear Prof Vijilant

Can we Design tunnels without a circular shape?.

Examples - Triangle, Square, Rectangle, Kite, Rhombus etc?.


Hello Prashant
Are you sure all tunnels have a circular shape?  They have probably been circular in section initially, because they were drilled. They are holes.  How do you make a hole in a piece of wood?  With a drill.
And that is how most tunnels are begun.  And then, of course, the bottom is filled in after burying cables etc.  But of course, a different drill could be used.  By using the idea of curves of constant width, a square hole could be drilled. Then a rectangular tunnel by two adjacent square tunnels. This was done by Watts Brothers Tool Works in Wilmerding, PA.  They make other drills, that drill sharp-cornered holes in the shape of pentagons, hexagons, and even octagons. Of course, it might be difficult to scale up to the size needed for a tunnel, but the possibility is there. And perhaps the strongest roof of a tunnel should be an inverted catenary.  I can't see any use for such shapes as a kite.
Best wishes

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