Here we show that the problem of DUPLICATING A CUBE cannot be solved using only straightedge and compass.

1. PROBLEM
THE PROBLEM OF DUPLICATING A CUBE CANNOT
BE SOLVED USING ONLY STRAIGHTEDGE AND
COMPASS
Kavi D. Pandya
SemII - 131020

2. Duplicating a cube
Existing cube : side ‘A’ and volume ‘A3’
New cube : side ‘xA’ and volume ‘2(A)3’
Determine x :
(xA)3 = 2(A)3
(x)3(A)3 = 2(A)3
x3 = 2
Therefore : x = 3√2
x = 3√2 = 1.2599210498948731647672106072782…..

3. Constructible numbers
DEFINITION: A real number ‘x’ is said to be constructible
by straightedge and compass if a segment of
length |x| can be obtained starting
from our unit segment by using a finite
sequence of straightedge and compass
construction.
x = 3√2 = 1.2599210498948731647672106072782…..
x is non-terminating number

4. Constructible Square roots
Proposition: Let ‘a’ be a constructible real number with
‘a’ > 0. Then, √a is constructible.
λ
λ = √a and is constructible

5. Constructible Number Theorem
Theorem: A number tєC is constructible if and only if
there exists an irreducible polynomial pєQ and
an integer j≥0 such that :
e.g. t2 – 4t = 0
t – 4 = 0 (No. 4 is constructible)
Similarly : t3 – 2 =0
(degree of t = 3)
t = 3 √2 is not-constructible ◊

6. Why only power of 2
The basic operations in the plane used in straightedge and
compass constructions are as follows:
(1) to draw a line through two given points
(2) to draw a circle with centre at a given point and radius equal
to the distance between two other given points
(3) to mark the point of intersection of two straight lines
(4) to mark the points of intersection of a straight line and a
circle
(5) to mark the points of intersection of two circles
Any straightedge and compass construction starts from given
points, lines, and circles and involves a finite sequence of steps of
these kinds to obtain some other points, lines, or circles.

7. Bibliography
1. Algebra Pure and Applied