what is an isometric isomorphism?
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Isomorphisms between groups are not the same thing as isomorphisms between, for example, rings.  There is a fully general definition of isomorphism that stems from universal algebra and mathematical logic, but I risk boring you if I develop it here.  So let me tackle the specific case of normed vector spaces, as I suspect it is in this setting that you came across the term "isometric isomorphism".\n\nTwo normed vector spaces are said to be isomorphic if there is a bijective map f between them that is homogeneous and linear, and for which there is a constant C>0 such that (1/C) ||x|| < ||f(x)|| < C ||x|| for all x.  An isometric isomorphism is an isomorphism for which that constant C can be taken to be 1, provided you reinterpret the symbol < as less than or equal--it's not easy to type the appropriate symbol.  In such a case, distances are not only roughly preserved (as in the case in which one has an isomorphism with constant C>1): they are indeed actually exactly preserved.  Thus the name isometry.
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Functions of the form (ax+b)/ (cx+d) are called homographic functions in mathematics. Homographic functions have two lines of symmetry: y=-d/c and x=a/c these lines intersect each other at A= (a/c,-d/c). A is point of symmetry.\n\nSimplest form of a homographic function is f=1/x where y=0 and x=0 (principle axes) are its lines of symmetry and A= (0, 0) (namely origin) is its point of symmetry.\n\nNow if your know the shape of f=1/x \n\nThen f=1/ (3x+2) is exactly the same but its point of symmetry is moved from origin to (1/3,-2/3).\n\nIf you don’t know the shape of 1/x refer to other answers above me or:\n////////////|/.\n////////////|//.\n////////////|//////.\n/_______________________.__ \n//./////////|\n///////.////|\n//////////./|\n\nignore slashes
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I would call it anterior to posterior/dorsal to ventral bilateral symmmetry - that is, if you drew a line through the animal from the front to back connecting its top and bottom, the resulting halves would be symmetrical.  However, I don't know if there are really specific types of bilateral symmetry - usually things are defined as asymmetrical, bilaterally symmetrical, or radially symmmetrical.
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S3 is isomorphic to the dihedral group of order 6, which consists of maps over the equilateral triangle with unique sides. There are only two basis elements, the flip over a central axis, and a single rotation. All of the other 4 maps can be written as products of these two actions. Since a homomorphism is completely defined by its action on the basis, we have 2! automorphisms (ways of mapping the basis into itself).
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Isomers \nDefinitions of isomers on the Web:\n\n    * A long-lived excited state of the nucleus. Arbitrarily defined in as the Table of Isotopes as having a half-life greater than 1 ms.\n      ie.lbl.gov/education/glossary/Glossary.htm\n\n    * Compounds with the same chemical formulas, but different structures, and therefore different chemical or physical properties.\n      xenon.che.ilstu.edu/genchemhelphomepage/glossary/i.html\n\n    * Molecules with identical molecular formulas but differing in the sequence of bonding or arrangement in space of their atoms, ie their structural formulas.\n      www.visionlearning.com/library/pop_glossary_term.php\n\n    * When two materials with different properties and/or structures share the same chemical formula, they are said to be isomers\n      www.uyseg.org/greener_industry/pages/glossary.htm\n\n    * compounds having the same atomic composition (constitution) but differing in their chemical structure. [They include: structural isomers (chain or positional), tautomeric isomers, and stereoisomers - including geometrical isomers, optical isomers and conformational isomers.]\n      www.cartage.org.lb/en/themes/Sciences/Chemistry/Organicchemistry/Common/Common.htm\n\n    * Different compounds that have the same molecular formula. All isomers fall into either of two categories - that of constitutional isomers or stereoisomers.\n      www.agsci.ubc.ca/fnh/courses/glossary.htm\n\n    * compounds that have the same numbers and kinds of atoms but that differ in the way the atoms are arranged.\n      www.nutrabio.com/Definitions/definitions_i.htm\n\n    * are two or more compounds with identical chemical composition with respect to mass, number, and type of atoms, and which differ only in the arrangement of these atoms in space.\n      www.springboard4health.com/notebook/dict_i.html\n\n    * Two or more chemical compounds having the same structure but different properties.\n      www.uoguelph.ca/GTI/urbanpst/glossf_m.htm\n\n    * are compounds whose molecules contain the same atoms but in different arrangements.\n      www.dac.neu.edu/physics/b.maheswaran/phy1121/data/glos/glossary.htm\n\n    * (noun) molecules that are formed from the same atoms in the same chemical linkages but have different three- dimensional- conformations\n      www.orgsites.com/fl/adjuvantdisease/_pgg9.php3\n\n    * In chemistry, isomers are molecules with the same chemical formula and often with the same kinds of bonds between atoms, but in which the atoms are arranged differently. Many isomers share similar if not identical properties in most chemical contexts.\n      en.wikipedia.org/wiki/Isomers
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Two objects which are mirror images of each other are called enantiomers
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In an equilateral traingle, all sides are equal and all three angles are 60 degrees.\n\nIn an isosceles triangle, two sides are equal and the two base angles of an isosceles triangle are equal to one another.\n\nThe other answer above refers to light reflected in a mirror or other reflecting surface/
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Here is a web site that gives a nice example that might be helpful. Unfortunately, I am not an expert here so i can't give a quick answer, even if one exists. I suspect that one might have a group that is isomorphic to a Galois group, and not know it.
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associative, commutative, identity, inverse
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equal to or exactly the same as
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Stationary in reference to what fixed point?
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i think its true\nim f is isomorphical  with G/(ker f) so\n|G| = |ker f|*|G/(ker f )| = |ker f| *|im f|.\n i applied two well known theorems.
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This is an easy problem.  Here are the rough ingredients:\n\n(1) f is an isometry (i.e. d(f(x),f(y)) = d(x,y) for all x,y in X) implies that f is continuous.  (Prove this!)\n\n(2) The image of a compact set under a continuous map is a compact set, and so f(X) is a compact subset of X.\n\n(3) Suppose, for contradiction, that f is not onto.  So there is x in X &#92; f(X).  Let epsilon be positive real number less than dist(x,f(X)) (this distance is defined to be the infimum of the distances d(x,y) for all y in f(X)).  Since X is compact, X can be cover by finitely many balls of radius epsilon.  Let N be the smallest number of balls of radius epsilon with which you are able to cover X.  x belongs to one of these balls, obviously, and such a ball is disjoint from f(X).  So there is a cover of f(X) consisting of at most N-1 balls of radius epsilon.  But the inverse images (under f) of these balls (all of them of radius epsilon because f is an isometry) cover X.  Since we chose N to be minimal, and we have found now an epsilon-cover of X consisting of N-1 balls, we have reached a contradiction.
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Rotational symmetry: if something is rotated a fixed set of degrees about an axis, and it remains the same, its called rotational symmetry. eg, circle rotated about its center.\n\nLine Symmetry: If an object is symmetrical about any given axis. eg, diagonal of a square.\n\nCongruence: Implies exact match. Two toyota cars can be thought to be congruent.\n\nSimilarity: Implies the angles at the vertices match, but dimensions don't match. eg an equilateral triangle of area 1cm^2 and 100cm^2 are similar, not congruent.
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symmetry makes reading the x-ray crystallograph much much easier.
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equal number of sides and equal measure of angles\n\n\nor at least thats what i remember from geometry!
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This is an inevitable occurrence; there is actually a theorem by Paul Erdos (I forget who he was working with) that is applicable...  Erdos-Schekeres Theorem, perhaps.  It's the one where they proved that if you take a piece of paper, cover the bottom of a box with it, remove it, wad it up and throw it back into the box, at least one point will be exactly above the same place it was before...  It doesn't take a whole lot of imagination to see a definite isomorphism between the E-S theorem (or whichever one it is) and what you're talking about.\n\nANYHOW, what it boils down to, in your case, is that you will have an experience (whether in dream or in reality is not germain) and it will match a future one.  On the whole, you should expect that to happen once in a while--and if you have a wide enough breadth of experience, perhaps more often...\n\nI'm not sure that this addresses your question.  If it doesn't, here's a shorter answer: It is part of normal life; it isn't a problem; and we all share it to some degree or other.