Compass and straightedge in the context of "Proof of impossibility"

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⭐ Core Definition: Compass and straightedge

In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a compass.

The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so it may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass; see compass equivalence theorem. Note however that whilst a non-collapsing compass held against a straightedge might seem to be equivalent to marking it, the neusis construction is still impermissible and this is what unmarked really means: see Markable rulers below.) More formally, the only permissible constructions are those granted by the first three postulates of Euclid's Elements.

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👉 Compass and straightedge in the context of Proof of impossibility

In mathematics, an impossibility theorem is a theorem that demonstrates a problem or general set of problems cannot be solved. These are also known as proofs of impossibility, negative proofs, or negative results. Impossibility theorems often resolve decades or centuries of work spent looking for a solution by proving there is no solution. Proving that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a proof that works in general, rather than to just show a particular example. Impossibility theorems are usually expressible as negative existential propositions or universal propositions in logic.

The irrationality of the square root of 2 is one of the oldest proofs of impossibility. It shows that it is impossible to express the square root of 2 as a ratio of two integers. Another consequential proof of impossibility was Ferdinand von Lindemann's proof in 1882, which showed that the problem of squaring the circle cannot be solved because the number π is transcendental (i.e., non-algebraic), and that only a subset of the algebraic numbers can be constructed by compass and straightedge. Two other classical problems—trisecting the general angle and doubling the cube—were also proved impossible in the 19th century, and all of these problems gave rise to research into more complicated mathematical structures.

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Compass and straightedge in the context of Squaring the circle

Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.

In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi () is a transcendental number.That is, is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found.

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Compass and straightedge in the context of Doubling the cube

Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related problems of squaring the circle and trisecting the angle, doubling the cube is now known to be impossible to construct by using only a compass and straightedge, but even in ancient times solutions were known that employed other methods.

According to Eutocius, Archytas was the first to solve the problem of doubling the cube (the so-called Delian problem) with an ingenious geometric construction. The nonexistence of a compass-and-straightedge solution was finally proven by Pierre Wantzel in 1837.

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Compass and straightedge in the context of Pierre Wantzel

Pierre Laurent Wantzel (5 June 1814 – 21 May 1848) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge.

In a paper from 1837, Wantzel proved that the problems of doubling the cube and trisecting the angle are impossible to solve if one uses only a compass and straightedge. In the same paper he also solved the problem of determining which regular polygons are constructible: a regular polygon is constructible if and only if the number of its sides is the product of a power of two and any number of distinct Fermat primes (i.e., the sufficient conditions given by Carl Friedrich Gauss are also necessary).

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LINKED FROM (PARENTS)
← Squaring the circle ← Doubling the cube ← Pierre Wantzel ← Proof of impossibility
LINKS TO (CHILDREN)
Hexagon → Geometry → Angle → Idealization (science philosophy) → Ruler → Compass (drawing tool) → Straightedge → Compass equivalence theorem → Neusis construction → Euclidean geometry → Euclid's Elements →