7/24/2019 Drilling Square Holes

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rilling Square Holes

rill bit that produces square holes it

defies

common

sense. How

can

a revolving

edge

cut anything but a circular

hole?

Not

do su

ch

drill bits exist, as well as bits for

pen-

hexago

nal

, and octagonal holes, but they

e their s

hap e

from

a simple geometric

con-

known as a Reu leaux triangle, which was

after Franz Reuleaux , 1829- 1905.

construct a Reuleaux triangle, start

with

an

r

al

triangle of

si

d

es

(fig.

.

Wi

th a radius

to s and the center at one of the vertices,

an arc connecting the other two vertices. Sim-

y, draw arcs connecting the en dpoin ts of the

two sides. The three arcs form the Reuleaux

gle. One of its properties is that of

constant

h mea

nin

g that the figure

could

be rotated

two parallel lines separated

by

nces and always be tangent to each.

Fig. 1

A Reuleaux triangle

is property of constant width introduced the

in

a sidebar of our geometry text

(M

oise

and

Downs 1982, 555

). This figure has

width, I lectu red, ju

st lik

e a

circle.

t thinking, I volunteered,

Ima

gine it

as

on

a cart. What sort

of

cart? a student

Wh

y, a math cart,

to

carry my board com-

protractor, I replied, digging m

yse

lf in

r. Thi s

was

the first

of

seueral

impulsive

mi

s-

I made about the Reuleaux triangle,

to admit

my

errors after a little reflection. Not

year of teaching had my intuition failed

so com

pl

etely.

e constant-width property can

be

used

to

loads, but not

by

us

in

g Reuleaux trian

as wheels. If several logs had congruent

triangles as cross sections, bulky items

86,

No.

7

October

1993

Fig. 2

Reuleaux logs

could be rolled on

top

of a base of

them

(fig. 2).

Movement would occur as logs were transferred

from back to fr

ont,

providing a movab le base of

con-

stant height.

t

has

been

proposed that the

Egyp-

tians moved the massive stone

blocks

for the Great

Pyramids

in

a

si

mil

ar manner.

Bu

t the

Reuleaux

tria

ngl

e cannot be a wheel.

The only

conceivable

point for the axle , at the cen-

ter of the triangle, is not the same distance from

the Reulea

ux

triangle's

sides

(fig. 3).

f

he sides

of the equilateral triangle ares, then applying the

property that the centroid is two-thirds the

di

s-

tance from a vertex to the

opposite si

des

gives

= ~

~ = ~ s

3 2 3

1)

0.577s,

whereas

0.423s.

Even if

four

Reuleaux-triangle wheels were syn-

chronized, the load would rise and

fall

continuous-

Scott Smith

is th

e comp

ut

er coordinator

at

University

School

Chagrin Falls

OH 44022

Heis

interested

in

tech

nology

that illuminates

mathematics

and

is an

author o

commercial software

Scott G. Smith

What is

the center o

aReuleaux

triangle

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How can

a euleaux

bit e

used

to

drill

a square

hole

Fig. 3

A

uleaux wheel

ly. You

would

need to take motion-sickness pills to

ride this

cart

"And since

it has constant

width,

it

would

just

fit

insi

de a square whose sides are that

width,"

I con

tinued, trying

to

regain

ud

ents' attention. I care

fully

drew a square circumscribing the Reuleaux

triangle fig 4). Two

of

the sides

of

the triangle are

tangent to two sides

of

the square and two vertices

of

the triangle intersect the square directly opposite

those points

of

tangency, as shown in figure 4a. If

the

figure

is turned as

seen in

figure

4b

,

one si

de

of

the

Reuleaux

triangle is tangent

to one side of

the square directly opposite

one of

the vertices

of

the triangle. All three vertices intersect sides

of

the

square.

"If

the Reuleaux triangle just

fits

inside the

square, no matter what

position

it's in, couldn't it

rotate around the inside

of

the

square?" The

stu

dents

needed

to be convinced;

a

model would have

to be built. But if it

i

rotate around the

inside

doesn't that

mean

that a sharp Reuleaux

t r i n g l ~

could

carve out a square as it rotated? I had them.

"Drill

a square hole?"

one

student countered. "No

way "

That night I cut a Reuleaux triangle with side 10

centimeters from a manila l

der to

take

to

class

the next day. With

a lot

of

effort, I

was able to show

that the triangle

could

rotate around the inside of a

ten -centimeter square.

And

ifthis shape

was made

Fig

4

A uleaux tnangle tnscribed wtthtn a square

Q

Fig. 5

A tracing of the path of the

Reuleaux

tr

iangle's centroid

of

metal

and

placed

at the

end

of

a rotating

shaft, it

would

cut out a square, I continued, racking

up

two

more falsehoods.

First, I implied that the

cen

ter of the Reuleaux triangle would

coincide

with the

center

of

a drill's shaft; it cannot.

And second,

the

corners

of

the holes are not right angles but

are

slightly

rounded.

(a)

Fig. 6

Reuleaux tria

le rotating within a square

THE

MATHEMATICS TEACHER

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to show that the triangle should be cen

at the end of a rotating shaft, I stuck a pen

triangle's center, and while a student

tJiangle within the square, I

the center's path on paper (fig. 5). It's defi

a single point

,"

I had

to

admit, holding up

raced curve,

"b

ut it sure looks like a circle "

4.

what

is

the path of the centroid of a Reul

triangle while it

is

boring a square hole? As-

that the square and the equilateral triangle

sides of length

1.

Center the square about the

po

sit

ion

the Reuleaux triangle

so

that

A

is at

1-1/2,

OJ, as in figure

6a.

Using

(1,

tiangle

's

centroid

will be

P(-1/2 +

..f313, O.

im

agine rotating the triangle

clockwise

through

o

sition

in

figure

6b,

ending

up in

figure 6c,

centroid is

P (O, -112

+ ..J3/3). The path

P to P lies in

quadrant I. In figure

6b

let

a

the counterclockwise angle

by

A'P'

and a horizontal line through

A;

and

they-coordinate ofpointA'. We are interested

in

of

P'.

Note that cos a= 1/2 +

c

and

270

a+ 30 = 300 + a.

Also

note that

this rotation

from

figures 6a through 6b ,

a

from 60 degrees

to 30

degrees. Si

nee AP'

=

if we measure

from

the coordinates of A'(-1/ 2,

x

andy coordinates of

P'

can be

found:

x= ;_

1

+

~

cos(300

+a

_ -3+ , 3

cos a+3

sin

a

- 6

y=

+

~

sin(300o

a

)

c o s a - ~ + t sin(300 +a

_ -3 + 3 cos a ,3 sin a

- 6

goes from

60

degrees

to

30 degrees. Finding

ath of the triangle's center in the other three

F1g 7

of Reuleaux triangle

centro1d

1ns1de

a Circle

86,

No.

7

October

1993

quadrants is similar in procedure and produces

equations that are symmetric to the

01igin

and both

axes.

Quadrant II:

Quadr

ant III:

Quadrant IV:

3 , 3 cos

a-3

sin a

X

= .:::.._.....:...=-=c::.......: ---...::.._;: -'-

6

-3+3 cos a+ \ 3

sin a

y=

6

x = 3

' 3 cos 3 sin

a

. 6

Y =3- 3 cos a - ' 3 sin a

6

x

=- 3 + \ 3 cos

a

3 sin

a

6

Y =3 - 3

cos a - ,

3

sin a

6

But these equations do not describe a circle. In

equations (2) and (3), when

a =

30, Pis on the

x-axis at approximately (0.07735 , 0). But

when

a =

45

,

X =y = 6

v

3\ 2 ,

which

mak es the distance

from P' to

the origin

about 0.081

68.

This noncircular situation is

also

shown by graphing the foregoing four parametric

equations with a circle whose radius is slightly

smaller or larger. In figure 7, the circle

is

the outer

curve. Note that the centroid's path is farther

from

the circle at the axes than at the midquadrant

point.

Th e Reul eaux triangle's centroid does not follow

a circular path. How then is the Reuleaux drill bit

contained within the square outline that it is

to

cut? Harry Watts designed a drill in 1914 with a

patented full floating chuck"

to

accommodate his

irregular bits. Bits for square, pentagonal,

hexago

nal , and octagonal

holes

continue to be sold by th e

Watts Brothers T

ool Works in

Wilmerding, Penn

sylvania. The actual drill bit

for

the square is a

Reuleaux triangle that is concave

in

three spots

to

allow

for

the

co

rners

to be

cut without shavings

obstructing the path (fig.

8).

Even the

modified

drill bit leaves slightly round

ed

corners. H

ow

rounded? Assume the starting

po

sition

in

figure 9a,

in

which the Reuleaux triangle

is just tangent at point C.

As

the triangle rotates

counterclockwise, C leaves that edge of the quare

temporarily (labeled

c n

fig. 9b) only

to

rejoin an

other edge at position

C"

in figure 9c. In figure 9b,

let

a

be

mLMA'B',

the angle

formed

by A'C'

and the horizontal line through

A',

and

c

be

the

y-coordinate of

A.

Then =

a +

60 - 90 =a

30

Bits exist

th t can cut

pentagonal

hexagonal

nd

octagonal

holes

581

7/24/2019 Drilling Square Holes

4/5

With this

information,

the teacher

can avoid

the blunders

I

m

e

2

Watts

chuck and

drill

Cross section of

dr

ill in hole

Fig.

8

B t

and chuck

that drill square holes

(From

ardner

[1963])

and

cos

a

1/2

+c. To

generate the

corner

by C,

a

starts at 30 degrees in figure 9a

and

ends up at 60

degrees in figure

9c. From

using A C = 1

and mea-

suring from the

coordinates

of

A, the coordinates

of

C are de

sc

ribed by

and

x

-

1

+ 1 cos(a

30

2

=

- 1+ \

os

+sin

2

y =

c

+

si

n(

30

= ( cos

a ~

) +sin(

a 30

)

=

- 1+ cos

a

+ \ 3 sin

a

2

Th

e equations for the other three corners a

re

simi-

lar and when graphed with the rest

of

the square

yie

ld gure

10

Not

on ly

does

the

Reul

ea

ux

tr

ian

gle have

practi-

ca

l

and in te

r

esti

ng

applicatio

ns

and is easy

to

B

(c)

Fig. 9

Reuleaux tri

angle

rotating within a square

d

escribe geometrically,

but it generates a

lot of

is

cu

ssion owing to

its nonintuitive

properties. With

this background , the teacher can

avoid

the blun-

ders I made. Further explorations into the topic

might include studying other figures

of

constant

width

(

see

Gardner [

1963

, Rademacher

and

Toeplitz [1990], and

Johnson [1

989 ] ; identifying

further the curve

of

the Reuleaux triangle s

center

as

it cuts a square;

and

noting the shape

of bit

s for

pentago

nal

, hexagonal,

and

octagonal

holes

.

Fig. 10

Path of vertex

of

Reuleaux triangle

rotating within a square

THE MATHEMATICS TEACHER

7/24/2019 Drilling Square Holes

5/5

BIBLIOGRAPHY

Mathematical

Games. Scientific

208

(February 1963):

148-56.

Millie.

4

What

Is a

Curve

ofConstant

Student

Math

Notes (March 1989):1-4.

Edwin, and Floyd Downs, Jr. Geometry. Read

Addison-Wesley Pu

bl

ishi

ng Co., 1982.

h

er,

Hans,

and Otto

Toeplitz.

The Enjoyment

athematics.

New

York: Dover P

ublications,

R

''Problem

Solving in Geometry-a

ofReuleaux Triangles. Mathematics

79 (January 1986):11-14.

Tool

Works.

How

to Drill

Square

,

, Octagon

Pentagon

Holes. Wilmerding,

Author

, 1966.

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