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Congruence and Similarity

Three red hearts, two identical and one about twice the size of the othersThe Roman historian, biographer and essayist, Plutarch (c.46-120 AD, formerly Greek Plutarkhos), once wrote of a Greek legend that raised an interesting philosophical question.  When Theseus, son of the mythical king of Athens, had returned from Crete after killing the Minotaur, his ship was preserved by the Athenians for many, many years.  Plutarch reports that, as older planks decayed, they were removed and replaced with stronger timbers.  This raised an interesting question among philosophers about whether the ship, once all its timbers had been replaced, could be considered the same ship, or not.

Many philosophical debates have been held over versions of this story.  Today for example, as people replace human body parts, it raises the question of when a person ceases to be the same person.  A famous twist on the story was produced by the English philosopher Thomas Hobbes (1588-1679) who wondered what would happen if the discarded planks were kept and assembled into a second ship … which ship (if either) could be considered the original ship of Theseus?

Mathematicians asked a similar question about shapes: “What can we change about something and still consider it to be the same thing?”  Obviously, changing the shape itself will obviously not be acceptable.  What about rotating or reflecting the shape?  What if it was simply moved (translated)?  Mathematicians deemed those three things to be acceptable.  Stretching and warping were not.  Nor were simple enlargements or reductions.

Mathematicians speak of shapes that retain their size (and shape) as being congruent.  Since we measure size with lengths and shape with angles, we summarise all this to say that shapes are congruent if one can be transformed into the other by an isometry (translation, rotations, or reflection) and that the properties of lengths/distances and angles are invariant.

You will see that the next topics on this website will play with these transformations (translation, rotations, reflection and others … the kind that any good graphics manipulation program will have).  Mathematicians like to search for symmetries, and then use them to create interlocking patterns (tessellations).

Similar shapes preserve all the angles (shapes) but allow enlargements or reductions in size.  When you watch television or view an image on your phone you are enjoying the benefits of similarity transformations.

History of the symbols and history of the definitions and understanding of transformations ... Galois?

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