How to Drill a Square Hole!

7/30/2019 How to Drill a Square Hole!

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How to Drill a Square Hole!

By Stephen Casey


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Imagine you are an Egyptian wishing to transport aheavy block.

Cylindrical logs motion is flat and horizontal, does notbob up and down.

Reason vertical cross-section of the cylinders arecircles.

Circles have constant width!


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Is the Circle Unique in This

Respect?

After thinking over this

problem for a while one

could come to the

conclusion that the circle

is unique in this respect.

This is incorrect!

In fact, there are

infinitely many such

curves.
http://www.miniwage.com/images/fun%20images/Thumbnails/Hal%20thinking.jpg


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The Reuleaux Triangle

Next to the circle the

reuleaux triangle is the

simplest such curve.

Shape formed by the

intersection of three

circles of equal radii,

each of which intersectsthe centres of the other

two.
http://images.google.ie/imgres?imgurl=http://upload.wikimedia.org/wikipedia/commons/c/cf/ReuleauxTriangle.png&imgrefurl=http://en.wikipedia.org/wiki/Reuleaux_triangle&h=200&w=200&sz=2&tbnid=zSO93j1dyXuH2M:&tbnh=99&tbnw=99&hl=en&start=7&prev=/images%3Fq%3Dreuleaux%2Btriangle%26svnum%3D10%26hl%3Den%26lr%3D%26sa%3DN


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Area of the Reuleaux Triangle

For a given width w, the areaof the reuleaux triangle is

given as follows:

=

This is the smallest area ofsuch a curve for a givenwidth, w.

23

2w

23

2 3w

2w
http://images.google.ie/imgres?imgurl=http://www.math.cornell.edu/~dtaimina/Reuleaux/triangle.gif&imgrefurl=http://www.math.cornell.edu/~dtaimina/Reuleaux/Reuleaux.htm&h=130&w=132&sz=4&tbnid=08PSmw7jIPQsTM:&tbnh=84&tbnw=86&hl=en&start=42&prev=/images%3Fq%3Dreuleaux%2Btriangle%26start%3D40%26svnum%3D10%26hl%3Den%26lr%3D%26sa%3DNhttp://images.google.ie/imgres?imgurl=http://www.math.cornell.edu/~dtaimina/Reuleaux/triangle.gif&imgrefurl=http://www.math.cornell.edu/~dtaimina/Reuleaux/Reuleaux.htm&h=130&w=132&sz=4&tbnid=08PSmw7jIPQsTM:&tbnh=84&tbnw=86&hl=en&start=42&prev=/images%3Fq%3Dreuleaux%2Btriangle%26start%3D40%26svnum%3D10%26hl%3Den%26lr%3D%26sa%3DN


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Constructing a Curve of Constant

Width

First construct a number ofequal straight lines forming a

closed path as shown. Place the compass at a given

vertex of the figure andconnect the two vertices towhich the vertex is joined byan arc.

Repeat, similarly for all othervertices of the graph.

The result is a curve ofconstant width of width equalto the length of the edges ofthe graph.


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Alternative method

Draw a number of arbitrary

straight lines which mutually

intersect each other. At the intersection of two such

lines place the compass and

draw an arc between the lines.

Then proceed around the

curve, connecting each arc tothe preceding one.

If this is done carefully enough

the curve will close and be of

constant width.


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The shape of the

reuleaux triangle can be

used to drill a (nearly)square hole.

We have the triangle

rotating about an axis

through its centre andthe axis itself tracing a

curve as shown.


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The geometric centroid

does not move in a

circle.

This path can be

approximated by a

superellipse with

equation:

1

r r

x y

a a


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Harry James Watts

In 1914, Harry James

watts invented a rotary

drill, based on thereuleaux triangle to drill

square holes.

Cross-section of the drill

is a reuleaux trianglemade concave in three

spots.


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Perimeter of the Reuleaux Triangle.

The length of an arc, l, ofa circle with radius w

subtending an angle qisgiven by the formula:

l = wq

In this case, q= /3 and

there are three such arcsof equal length hencethe perimeterp is givenby the formula:

p = w


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Perimeter of curves of constant

width

You may have noticed that for a given width

the perimeter of the Reuleaux Triangle is thesame as that of the circle.

Surprisingly, for any curve of constant width w,

the perimeter of the curve is w.

Egyptian problem!!


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Curves of Constant width

Every curve of constant widthis uniquely defined by half ofits perimeter.

Once half of the curve isconstructed we can drawlines which are parallel to thetangents of the circle at aperpendicular distance w

away. This also provides another

way of constructing curves ofconstant width.


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Rotors

Any convex figure that can be rotated inside a

polygon or polyhedron while at all timestouching every side or face.

Each curve of constant width is a rotor for the

square.

The Reuleaux triangle is the rotor of least areafor a square.


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Kakeya Needle Problem

Closely related to the problem of rotors is the

Kakeya needle problem. What is the plane figure of least area in which

a line segment of length 1 can be rotated?

No answer Abram Samoilovitch Besicovitch.

Area can be made as small as we please and

line can be rotated.


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Take-home Problem!

What is the smallest convex area in which a

line of unit length can be fully rotated.

How to Drill a Square Hole!

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