compass and straightedge construction

compass and straightedge construction

In order to define what a

compass and straightedge construction

is, some preliminary

definitions

are :

•

A

compass

is a tool which can be used for drawing circles, or arcs thereof, whose radii are sufficiently small, and for measuring lengths.

•

A

straightedge

is a tool which can be used for drawing lines, or

segments

(

http://

planetmath

.org/LineSegment

) thereof.

Some people use the word

ruler

to refer to the tool that is called a straightedge here. This can cause some confusion, however, because outside of mathematics, a ruler is used to measure any length (

http://planetmath.org/BasicLength

) desired. This is

not

a permissible tool for compass and straightedge constructions.

With these preliminaries out of the way, we can now proceed to the main definition.

A

compass and straightedge construction

is the provable creation of a geometric figure on the

Euclidean plane

(or

complex plane

) such that the figure is created using only a compass, a straightedge, and specified geometric figures.

Typically, if no preexisting geometric figure is specified, the tacit assumption is that one can use a

line segment

of length

1

. Moreover, in such

instances

, one can specify what length represents

1

, but it must remain

constant

throughout the construction.

A geometric figure is

constructible

if it can be made from a compass and straightedge construction.

One has to be very careful with the terminology associated with compass and straightedge constructions. For example, the phrase “A

20

∘

angle is not constructible with compass and straightedge” refers to the fact that a compass and straightedge construction of a

20

∘

angle is not possible using the tacit assumption as described above. As another example, the following is an erroneous

argument

regarding compass and straightedge constructions:

A line segment of length

π

is constructible because, given a line segment of length

1

, I can extend it as a ray. Then I can measure the

distance

between the two

endpoints

with my compass. After that, I can open the compass

π

times wider. Finally, I can mark that distance on the ray from one of the endpoints of the original line segment.

The above argument is one of many reasons why the word

provable

appears in the definition. One can open the compass however wide one wants, and one can mark arcs and points however one wants, but the construction is

invalid

unless one can prove indisputably that the construction really is what is stated. In the above example, one cannot prove that the compass was opened

exactly

π

times wider than the length of the original line segment.

Compass and straightedge constructions are of historical significance. The ancient Greeks are the most well-known civilization for investigating these constructions on an

elementary

level. It should be pointed out that the compasses that they used were

collapsible

. That is, you could open the compass and draw an arc, but immediately after you removed a point of the compass from the plane where you drew the arc, the compass would close completely. It turns out that whether a collapsible compass or a modern-day compass is used to perform these constructions makes no

difference

. This statement is justified by the fact that one can use a collapsible compass to construct a circle with a given radius at

any

point as shown by this entry (

http://planetmath.org/circlewithgivencenterandgivenradius

).

One of the greatest applications of abstract algebra is being able to determine which constructions are possible and which are not. The between constructions and abstract algebra is that the set of all constructible points is in

one-to-one correspondence

with the elements of the field of constructible numbers. Without abstract algebra, one would be hard pressed to prove statements about constructibility such as “A

20

∘

angle is not constructible with compass and straightedge.”

Title

compass and straightedge construction

Canonical

name

CompassAndStraightedgeConstruction

Date of creation

2016-03-14 20:38:32

Last modified on

2016-03-14 20:38:32

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

23

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 01A20

Classification

msc 51M15

Synonym

straightedge and compass construction

Synonym

ruler and compass construction

Synonym

compass and ruler construction

Related topic

ConstructibleNumbers

Related topic

TheoremOnConstructibleAngles

Related topic

MotivationOfDefinitionOfConstructibleNumbers

Related topic

ClassicalProblemsOfConstructibility

Defines

compass

Defines

straightedge

Defines

ruler

Defines

constructible

Defines

collapsible compass

Defines

collapsible

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