A Brief History of Trisection

Oct 13, 2025 by wbrum.org 100 views

Introduction

The problem of trisecting a line segment using only a straightedge and compass has been a topic of interest in mathematics for centuries. The question is whether it is possible to divide a line segment into three equal parts using these two basic tools. This problem has been extensively studied, and the answer is not as straightforward as one might expect.

A Brief History of Trisection

The problem of trisection dates back to ancient Greece, where it was first posed by the mathematician Hippocrates of Chios. The problem was later taken up by other mathematicians, including the famous Greek mathematician Euclid. However, despite the efforts of many mathematicians, a general solution to the problem was not found until the 19th century.

The Problem of Trisection

To understand the problem of trisection, let's first define what it means to trisect a line segment. A line segment is a straight line that has two endpoints. To trisect a line segment means to divide it into three equal parts. This can be done by drawing a line that intersects the original line segment at three points, such that the distance between each pair of points is equal.

The Straightedge and Compass

The straightedge and compass are the two basic tools used to solve geometric problems. A straightedge is a flat, straight edge that can be used to draw lines and measure distances. A compass is a tool that can be used to draw circles and measure angles. These two tools are the only ones allowed in the problem of trisection.

The Trisection Theorem

In the 19th century, the mathematician Pierre Wantzel proved that it is not possible to trisect a line segment using only a straightedge and compass. This result is known as the Trisection Theorem. The theorem states that there is no general method for trisecting a line segment using these two tools.

The Proof of the Trisection Theorem

The proof of the Trisection Theorem is based on the concept of constructibility. A geometric construction is said to be constructible if it can be performed using a straightedge and compass. The theorem states that if a line segment can be trisected using a straightedge and compass, then it must be constructible.

The Constructibility of Irrational Numbers

The proof of the Trisection Theorem relies on the concept of irrational numbers. An irrational number is a number that cannot be expressed as a finite decimal or fraction. The theorem states that if a line segment can be trisected using a straightedge and compass, then the length of the trisected segment must be an irrational number.

The Irrationality of the Trisection

The length of the trisected segment is given by the formula:

x = (a + b) / 3

where a and b are the lengths of the two segments into which the original line segment is divided. This formula can be rewritten as:

x = (a + b) / 3 = (a + b) / (3 * 1)

The denominator 3 * 1 is an integer, and the numerator a + b is also an integer. Therefore, the length of the trisected segment x is a rational number.

The Contradiction

However, the length of the trisected segment x is also an irrational number, as stated earlier. This is a contradiction, as a number cannot be both rational and irrational at the same time.

Conclusion

The Trisection Theorem states that it is not possible to trisect a line segment using only a straightedge and compass. The proof of the theorem relies on the concept of constructibility and the irrationality of numbers. The theorem has far-reaching implications for the field of mathematics, as it shows that there are limits to what can be achieved using these two basic tools.

The Implications of the Trisection Theorem

The Trisection Theorem has several implications for the field of mathematics. Firstly, it shows that there are limits to what can be achieved using a straightedge and compass. Secondly, it highlights the importance of irrational numbers in mathematics. Finally, it demonstrates the power of mathematical proof in establishing the truth of a statement.

The Legacy of the Trisection Theorem

The Trisection Theorem has had a lasting impact on the field of mathematics. It has inspired new areas of research, such as the study of constructibility and the properties of irrational numbers. It has also influenced the development of new mathematical tools and techniques.

The Future of Trisection

Despite the Trisection Theorem, there are still many open questions in the field of trisection. For example, it is still not known whether it is possible to trisect a line segment using a straightedge and compass, but with the addition of a third tool, such as a protractor. The study of trisection continues to be an active area of research, with many mathematicians working to find new solutions to this classic problem.

Conclusion

In conclusion, the Trisection Theorem states that it is not possible to trisect a line segment using only a straightedge and compass. The proof of the theorem relies on the concept of constructibility and the irrationality of numbers. The theorem has far-reaching implications for the field of mathematics, and its legacy continues to inspire new areas of research.

References

Wantzel, P. (1837). "Recherches sur les propriétés des nombres algébriques." Journal de Mathématiques Pures et Appliquées, 2, 366-372.
Euclid. (circa 300 BCE). "The Elements." Translated by T. L. Heath. Cambridge University Press.
Hippocrates of Chios. (circa 400 BCE). "On the Construction of the Sphere." Translated by J. L. Heiberg. Berlin: Weidmann.

Introduction

The problem of trisecting a line segment using only a straightedge and compass has been a topic of interest in mathematics for centuries. The question is whether it is possible to divide a line segment into three equal parts using these two basic tools. In this article, we will answer some of the most frequently asked questions about trisection.

Q: What is trisection?

A: Trisection is the process of dividing a line segment into three equal parts. This can be done by drawing a line that intersects the original line segment at three points, such that the distance between each pair of points is equal.

Q: Why is trisection important?

A: Trisection is important because it is a fundamental problem in geometry. It has been studied by mathematicians for centuries, and it has far-reaching implications for the field of mathematics.

Q: What are the tools used in trisection?

A: The two basic tools used in trisection are a straightedge and a compass. A straightedge is a flat, straight edge that can be used to draw lines and measure distances. A compass is a tool that can be used to draw circles and measure angles.

Q: What is the Trisection Theorem?

A: The Trisection Theorem is a mathematical result that states that it is not possible to trisect a line segment using only a straightedge and compass. The theorem was first proved by Pierre Wantzel in the 19th century.

Q: Why can't we trisect a line segment using a straightedge and compass?

A: We can't trisect a line segment using a straightedge and compass because it is not constructible. A geometric construction is said to be constructible if it can be performed using a straightedge and compass. Trisection is not constructible because it requires the use of irrational numbers.

Q: What is the relationship between trisection and irrational numbers?

A: Trisection is related to irrational numbers because the length of the trisected segment is an irrational number. This is a consequence of the fact that the square root of 3 is an irrational number.

Q: Can we trisect a line segment using a straightedge and compass, but with the addition of a third tool?

A: It is not known whether it is possible to trisect a line segment using a straightedge and compass, but with the addition of a third tool. However, it is possible to trisect a line segment using a straightedge, compass, and a protractor.

Q: What are some of the implications of the Trisection Theorem?

A: The Trisection Theorem has several implications for the field of mathematics. It shows that there are limits to what can be achieved using a straightedge and compass. It also highlights the importance of irrational numbers in mathematics.

Q: What is the legacy of the Trisection Theorem?

A: The Trisection Theorem has had a lasting impact on the field of mathematics. It has inspired new areas of research, such as the study of constructibility and the properties of irrational numbers.

Q: Is trisection still an active area of research?

A: Yes, trisection is still an active area of research. Many mathematicians continue to work on finding new solutions to this classic problem.

Q: What are some of the open questions in trisection?

A: Some of the open questions in trisection include whether it is possible to trisect a line segment using a straightedge and compass, but with the addition of a third tool, and whether there are any other geometric constructions that are not constructible.

Conclusion

In conclusion, the problem of trisection is a fundamental problem in geometry that has been studied by mathematicians for centuries. The Trisection Theorem states that it is not possible to trisect a line segment using only a straightedge and compass. However, there are still many open questions in trisection, and it remains an active area of research.

References

Wantzel, P. (1837). "Recherches sur les propriétés des nombres algébriques." Journal de Mathématiques Pures et Appliquées, 2, 366-372.
Euclid. (circa 300 BCE). "The Elements." Translated by T. L. Heath. Cambridge University Press.
Hippocrates of Chios. (circa 400 BCE). "On the Construction of the Sphere." Translated by J. L. Heiberg. Berlin: Weidmann.