Download E1-9

The Three Geometric Problems
Modern day geometry designing methods have evolved from our traditional compass and
straightedge tools, yet all of our current design machines are based on the fundamental concepts
from these Euclidean tools. Currently, an individual can design hundreds if not thousands of
diverse and distinct shapes using digital resources or unique machines. Although the change has
benefited Mathematicians world wide, there are still riddles that “haunt” the math community.
They are referred to as the “Three Geometric Problems3.” They are as follows: (1) Squaring the
circle: one must construct a square equal in area to a given circle (2) Doubling the cube: one
must construct the side of a cube whose volume is double that of a cube of a given edge (3)
Trisecting an angle: one must attempt to trisect any given angle. With the advances of modern
technology, one could be deceived into thinking that these issues must have been solved by now,
correct? Wrong. The digital factor causes individuals to believe that the three problems are no
longer factors when creating diagrams relevant to them, however each of the problems presents
its challenges when they must be performed by man, instead of machine. The trial and error of
these problems led to important discoveries of mathematical concepts.
Before the amateur Mathematicians dive into solving these problems, we must address
the history behind each of the problems, and two unique theorems that can be applied to each
case They are as follows1:
Theorem A: The magnitude of any length constructible with Euclidean tools from a given unit
length is an algebraic number.
Theorem B: From a given unit length it is impossible to construct with Euclidean tools a segment
the magnitude of whose length is a root of a cubic equation having rational coefficients but no
rational root.
The first problem: squaring the circle. This can be interpreted as creating a square that
has the exact same area as a circle. “The first Greek to be associated with this problem was
Anaxagoras, who worked on it while in prison3.” His obsession with the circle began with a
fascination with the sun and a prideful attempt to convince his enemies and captors that each
planet was made of stone and the sun was a red hot stone. Although he may have been more than
likely delusional, his curiosity did bring forth the first few failures of the problem. The failures
were able to confirm the number categories at the time: constructible and transcendental. Several
methods were used in order to prove that the riddle could be solved. From Theorem A1 above,
individuals attempted to draw straight lines between two points, draw a circle with a given center
and radius, and finally find the intersection of two points/lines/circles. This proved unsuccessful
in the constructible sense. Consequently, it led to the discovery of transcendental2 numbers. The
most famous of these values being pi π that was discovered in the 19th century. “By not being
algebraic, the value is automatically constructible. However, the area of a circle with radius one
is exactly π. If a square with such area were constructible, so would the number itself2.” It was
through this conclusion that Mathematicians began to quantify other transcendental numbers that
could be used to explain values and terminology that could not be classified as constructible.
The next problem featured the doubling of a cube. In theory, an individual would
multiply the side of cube by a scalar of two, thus the volume of a cube would be doubled. This
problem originated from the island of Delos where the people were suffering from a plague3. The
cure to the plague was to double the size of the alter, however the people soon realized that
doubling the sides would not double the volume. Quite often referred to as the “Delian problem,”
this case was solved by Menaechmus3 in 350 B.C. using the original Euclidean tools and other
resources. The problem changes when analyzing from an algebraic perspective1. “We let x be the
edge of the cube having twice the volume of the given cube. Then x3-2=0.” It is evident that this
equation has no rational roots. Referring back to Theorem B, we note that x is not constructible.
This problem introduced complex roots. Although it is not a traditional scalar, there does exist a
value that makes the equation true, however it is not a real solution. Once again, the problem
produced a new method to solve a mathematical situation in a non traditional light.
Lastly, the problem dealing with trisecting an angle. This occurs when an individual
attempts to divide an angle into three equal parts. By intuitive thinking, one can tell that this only
works when the angle is a multiple of three, therefore, if an angle is not divisible by three, then
the problem is impossible. Many individuals attempted this with little to no success with the
original Euclidean tools, however, from the lecture notes3, there does exist a proof that solves the
problem. This was accomplished through Hippias who created the quadratrix curve. This was not
the only immediate solution to this dilemma. In fact, there came a point in time where
Mathematicians were more concerned about not being able to trisect a sixty-degree angle. “The
problem had been settled in 1837 by Pierre Laurent Wantzel who had proved that there was no
way to trisect a sixty-degree angle in the classical framework2.” An analysis from the perspective
of Theorem B illustrates that that x is not constructible given that x is one of the lengths of the
legs of the triangle from the angle formed. Despite this problem being solved, it led to a broader
style of mathematical proofs.
Overall, one of the most unique attributes to the three problems is that they are not solved
using geometry, rather, algebraic theorems and proofs. Consequently, “the problem stems from
the fact that all three problems are unsolvable with Euclidean tools; similarly, it took over 2,000
years to prove so.1” Ironically, it was the failure to solve these problems that led to some of the
most important mathematical concepts that are still used by Mathematicians today.
1. http://web.mst.edu/~lmhall/5-Three%20Problems.pdf
2. http://www.cut-the-knot.org/arithmetic/antiquity.shtml
3. https://elearning.uh.edu/bbcswebdav/pid-3039176-dt-content-rid20829173_1/courses/H_20163_MATH_4388_18577/h-6.pdf