Definition and Tessellation

Here is an explanation of how definitions are usually given by prescise writers. I got it from http://en.wikipedia.org/wiki/Definition

Genus-differentia definition

Rules of definition

1. A definition must set out the essential attributes of the thing defined.
2. Definitions should avoid circularity. To define a horse as 'a member of the species equus' would convey no information whatsoever. For this reason, Locke adds that a definition of a term must not consist of terms which are synonymous with it. This error is known as circulus in definiendo. Note, however, that it is acceptable to define two relative terms in respect of each other. Clearly, we cannot define 'antecedent' without using the term 'consequent', nor conversely.
3. The definition must not be too wide or too narrow. It must be applicable to everything to which the defined term applies (i.e. not miss anything out), and to no other objects (i.e. not include any things to which the defined term would not truly apply).
4. The definition must not be obscure. The purpose of a definition is to explain the meaning of a term which may be obscure or difficult, by the use of terms that are commonly understood and whose meaning is clear. The violation of this rule is known by the Latin term obscurum per obscurius. However, sometimes scientific and philosophical terms are difficult to define without obscurity. (See the definition of Free will in Wikipedia, for instance).
5. A definition should not be negative where it can be positive. We should not define 'wisdom' as the absence of folly, or a healthy thing as whatever is not sick. Sometimes this is unavoidable, however. We cannot define a point except as 'something with no parts', nor blindness except as 'the absence of sight in a creature that is normally sighted'.
6. In class I said "Never say 'is when'". The reason is that, in mathematics, the genus is never a time. Of course, one can use 'is when' in grammatical English as in "Dawn is when the birds sing loudest.".
7. A definition is unintelligble to a reader who does not know what the genus in the definition is. In mathematics this means that some terms must be left undefined. These terms are defined axiomatically, by specifying the laws (axioms) which they satisfy. In geometry books, the terms 'point' and 'line' (and a few others) are usually left undefined in this sense. It was only in 1928 that mathematicians realized that all mathematics could be based on a single undefined notion, namely the notion of 'set'.

Tesselations

1. A tessellation of the plane is a collection of polygonal regions (called tiles) whose union is the entire plane and whose interiors do not intersect. A tessellation is said to be well aligned if any two regions meet either in a common edge or a common vertex or not at all. A tessellation whose tiles are all congruent regular polygons is called regular.
2. A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. -- http://en.wikipedia.org/wiki/Tessellation
3. • Easier - A tessellation is created when a shape is repeated over and over again. All the figures fit onto a flat surface exactly together without any gaps or overlaps.
   • Harder - A tessellation is a repeating pattern composed of interlocking shapes (usually polygons) that can be extended infinitely. The tiling for a regular (or periodic) tessellation is done with one repeated congruent regular polygon covering a plane in a repeating pattern without any openings or overlaps. Remember 'regular' means the sides of the polygon are all the same length, and 'congruent' means that the polygons fitted together are all the same size and shape. A semi-regular (or non-periodic) tessellation is formed by a regular arrangement of polygons, identically arranged at every vertex point.
   -- http://42explore.com/teslatn.htm
4. A tessellation is a regular tiling of polygons (in two dimensions), polyhedra (three dimensions), or polytopes (n dimensions). -- http://mathworld.wolfram.com/Tessellation.html
5. In geometrical terminology a tessellation is a pattern resulting from the arrangement of regular polygons to cover a plane without any interstices (gaps) or overlapping. The patterns are usually repeating. Regular tessellations are made up entirely of congruent regular polygons all meeting vertex to vertex. -- http://en.mimi.hu/gis/tessellation.html

Assignment (Justify your answers.)

1. In each of the above definitions of 'tessellation' identify the genus.
2. Which of the above definitions of 'tessellation' are equivalent (i.e. delineate exactly the same concept)?
3. For any two definitions above which are not equivalent, give an example satisfying one but not the other. (you may refer to the examples on pages 102 and 103 of EGT.)