Approximation Challenge

The Three Problems of Antiquity

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Date:

The Greek Constraint

Euclidean geometry is defined by the use of only two tools: an unmarked straightedge and a collapsible compass. In this activity, you will attempt to approximate the three "impossible" solutions. Observe where the standard tools fail to provide an exact geometric logic.

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Task 1: The Angle Trisection

Given the angle \(\angle ABC = 60^\circ\) below, attempt to divide it into exactly three equal parts of \(20^\circ\) using only your straightedge and compass. Describe your strategy below.

Observation: At what point did you find yourself needing to "eyeball" or "guess" the placement?

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Task 2: Doubling the Cube (Delian Problem)

Given a line segment of length \(s\) representing the side of a cube, you need to construct a segment of length \(x\) such that \(x^3 = 2s^3\). This implies \(x = s\sqrt[3]{2}\). Try to find this length geometrically.

Volume = V

Construct side \(x\) here

Challenge: You can construct \(\sqrt{2}\) using a right triangle. Why is \(\sqrt[3]{2}\) fundamentally different for a compass?

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Task 3: Squaring the Circle

Given a circle of radius \(r=1\), the area is \(\pi\). To construct a square of equal area, you must construct a side length of \(L = \sqrt{\pi} \approx 1.772\). Use the space below to attempt a construction that "looks" correct.

r=1

Area = \(\pi\)

The Transcendental Wall

Algebraically, squaring the circle is impossible because \(\pi\) is not algebraic—it is transcendental. It is not the root of any polynomial with rational coefficients.

Axiom Check: Compass + Straightedge = Quadratic Extensions Only

The Three Classical Problems

Impossibility Introduction Slides

Approximation Challenge Worksheet

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