Principle in compass and straightedge constructions
In geometry, the compass equivalence theorem is an important statement in compass and straightedge constructions. The tool advocated by Plato in these constructions is a divider or collapsing compass, that is, a compass that "collapses" whenever it is lifted from a page, so that it may not be directly used to transfer distances. The modern compass with its fixable aperture can be used to transfer distances directly and so appears to be a more powerful instrument. However, the compass equivalence theorem states that any construction via a "modern compass" may be attained with a collapsing compass. This can be shown by establishing that with a collapsing compass, given a circle in the plane, it is possible to construct another circle of equal radius, centered at any given point on the plane. This theorem is Proposition II of Book I of Euclid's Elements. The proof of this theorem has had a chequered history.[1]
^Toussaint, Godfried T. (January 1993). "A New Look at Euclid's Second Proposition" (PDF). The Mathematical Intelligencer. 15 (3). Springer US: 12–24. doi:10.1007/bf03024252. eISSN 1866-7414. ISSN 0343-6993. S2CID 26811463.
and 23 Related for: Compass equivalence theorem information
In geometry, the compassequivalencetheorem is an important statement in compass and straightedge constructions. The tool advocated by Plato in these...
transferred even with a collapsing compass; see compassequivalencetheorem. Note however that whilst a non-collapsing compass held against a straightedge might...
equivalence between the algebraic and geometric definitions of constructible numbers has the effect of transforming geometric questions about compass...
steps with compass and straightedge. In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves...
requirements of a valuation. According to Ostrowski's theorem, up to a natural notion of equivalence, the real numbers and p {\displaystyle p} -adic numbers...
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...
intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier...
Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal...
This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem. Nash embedding theorems. They...
in solving geometrical problems that could not be solved using standard compass and straightedge construction. These curves include: The conic sections...
theory. Wiles' proof of the longstanding conjecture called Fermat's Last Theorem is an example of the power of this approach. In classical algebraic geometry...
circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs...
pair having the same length and orientation. This application of an equivalence relation dates from Giusto Bellavitis's introduction of the concept of...
known assertions. A theorem of such a theory is either an axiom or an assertion that can be obtained from previously known theorems by the application...
ozone layer from the solar wind and is important in navigation using a compass. The force on an electric charge depends on its location, speed, and direction;...
one of the more recent terminologies, to "formal equivalence"; and "paraphrase", to "dynamic equivalence". Strictly speaking, the concept of metaphrase—of...
D(\mathbf {P} ^{\vee ,d}).} The main theorem about this transform is that this transform induces an equivalence of the categories of perverse sheaves...
Thomson. He had extensive maritime interests and worked on the mariner's compass, which previously had limited reliability. Kelvin was ennobled in 1892...
then AC is parallel to A'C'. The affine concept of parallelism forms an equivalence relation on lines. Since the axioms of ordered geometry as presented...
on surfaces, and predicting Einstein's fundamental observation of the equivalence principle a full 60 years before it appeared in the scientific literature...
Global Information