Mohr–Mascheroni theorem

In mathematics, the Mohr–Mascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone. The result was originally published by Georg Mohr in 1672,[1] but his proof languished in obscurity until 1928.[2][3] The theorem was independently discovered by Lorenzo Mascheroni in 1797.[4]

Proof approach

To prove the theorem, each of the basic constructions of compass and straightedge need to be proven to be doable by compass alone. These are:

  1. Creating the line through two existing points
  2. Creating the circle through one point with centre another point
  3. Creating the point which is the intersection of two existing, non-parallel lines
  4. Creating the one or two points in the intersection of a line and a circle (if they intersect)
  5. Creating the one or two points in the intersection of two circles (if they intersect).

Since lines cannot be drawn without a straightedge (1.), a line is considered to be given by two points. 2. and 5. are directly doable with a compass. Thus there need to be constructions only for 3. and 4.[5]

An alternative algebraic approach uses the isomorphism between the Euclidean Plane and . This approach can be used to provide significantly stronger versions of the theorem.[6] It also clearly shows the dependence of the theorem on Archimedes' axiom (which cannot be formulated in a first-order language).

See also

Notes

  1. ↑ Georg Mohr, Euclides Danicus (Amsterdam: Jacob van Velsen, 1672).
  2. ↑ Hjelmslev, J. (1928) "Om et af den danske matematiker Georg Mohr udgivet skrift Euclides Danicus, udkommet i Amsterdam i 1672" [Of a memoir Euclides Danicus published by the Danish mathematician Georg Mohr in 1672 in Amsterdam], Matematisk Tidsskrift B , pages 1–7.
  3. ↑ Schogt, J. H. (1938) "Om Georg Mohr's Euclides Danicus," Matematisk Tidsskrift A , pages 34–36.
  4. ↑ Lorenzo Mascheroni, La Geometria del Compasso (Pavia: Pietro Galeazzi, 1797). 1901 edition.
  5. ↑ Norbert Hungerbühler, "A Short Elementary Proof of the Mohr–Mascheroni Theorem," The American Mathematical Monthly, vol. 101, no. 8, p. 784, Oct. 1994.
  6. ↑ Arnon Avron, "On strict strong constructibility with a compass alone", Journal of Geometry (1990) 38: 12.

External links


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