Interesting Esoterica

Comparison of geometric figures

Article by Glenis, Spyros and Kapovich, M. and Brodskiy, N. and Dydak, J. and Lang, U. and Ballinger, B. and Blekherman, G. and Cohn, H. and Giansiracusa, N. and Kelly, E. and Others
  • Published in 2008
  • Added on
In the collection
Although the geometric equality of figures has already been studied thoroughly, little work has been done about the comparison of unequal figures. We are used to compare only similar figures but would it be meaningful to compare non similar ones? In this paper we attempt to build a context where it is possible to compare even non similar figures. Adopting Klein's view for the Euclidean Geometry, we defined a relation "<=" as: S<=T whenever there is a rigid motion f so that f(S) is a subset of T. This relation is not an order because there are figures (subsets of the plane) so that S<=T, T<=S and S, T not geometrically equal. Our goal is to avoid this paradox and to track down non-trivial classes of figures where the relation "<=" becomes, at least, a partial order. Such a class will be called a good class of figures. A reasonable question is whether the figures forming a good class have certain properties and whether the algebra of these figures is also a good class. Therefore we classified the figures into those that cause the paradox mentioned above and those that never cause it. The last ones are called good figures. Although simple, the definition of the good figure was difficult to handle, therefore we introduced a more technical, but intrinsic and handy definition, that of the strongly good figure. With these tools we constructed a new context, where we expanded our perspective about the geometric comparison not only in the Euclidean but also in the Hyperbolic and in the Elliptic Geometry. Eventually, there are still some open and quite challenging issues, which we present them at the last part of the paper.

Comment

Once you've said that two shapes aren't similar, you can be more precise and define a partial order that puts nearly-the-same figures closer together.

Links

Other information

isbn
7774553983
issn
1551-3440
journal
the montana mathematics enthusiast
keywords
euclidean geometry,isometries,klein,relations
number
2&3
pages
199--214
publisher
Iap
volume
5
arxivId
0611.062

BibTeX entry

@article{Glenis2008,
	title = {Comparison of geometric figures},
	author = {Glenis, Spyros and Kapovich, M. and Brodskiy, N. and Dydak, J. and Lang, U. and Ballinger, B. and Blekherman, G. and Cohn, H. and Giansiracusa, N. and Kelly, E. and Others},
	url = {https://arxiv.org/abs/math/0611062 https://arxiv.org/pdf/math/0611062},
	urldate = {2011-02-03},
	year = 2008,
	abstract = {Although the geometric equality of figures has already been studied thoroughly, little work has been done about the comparison of unequal figures. We are used to compare only similar figures but would it be meaningful to compare non similar ones? In this paper we attempt to build a context where it is possible to compare even non similar figures. Adopting Klein's view for the Euclidean Geometry, we defined a relation "<=" as: S<=T whenever there is a rigid motion f so that f(S) is a subset of T. This relation is not an order because there are figures (subsets of the plane) so that S<=T, T<=S and S, T not geometrically equal. Our goal is to avoid this paradox and to track down non-trivial classes of figures where the relation "<=" becomes, at least, a partial order. Such a class will be called a good class of figures. A reasonable question is whether the figures forming a good class have certain properties and whether the algebra of these figures is also a good class. Therefore we classified the figures into those that cause the paradox mentioned above and those that never cause it. The last ones are called good figures. Although simple, the definition of the good figure was difficult to handle, therefore we introduced a more technical, but intrinsic and handy definition, that of the strongly good figure. With these tools we constructed a new context, where we expanded our perspective about the geometric comparison not only in the Euclidean but also in the Hyperbolic and in the Elliptic Geometry. Eventually, there are still some open and quite challenging issues, which we present them at the last part of the paper. },
	comment = {Once you've said that two shapes aren't similar, you can be more precise and define a partial order that puts nearly-the-same figures closer together.},
	isbn = 7774553983,
	issn = {1551-3440},
	journal = {the montana mathematics enthusiast},
	keywords = {euclidean geometry,isometries,klein,relations},
	number = {2{\&}3},
	pages = {199--214},
	publisher = {Iap},
	volume = 5,
	arxivId = {0611.062},
	collections = {geometry}
}