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

The 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

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?

Our son Joshua is currently in year 11 preparing for the HSC. Josh was doing reasonably well in mathematics and yet we felt, and his teachers agreed, that if he really applied himself he could be achieving much better results. We asked around for a maths tutor and Graeme Henderson came highly recommended. Joshua began to see Mr Henderson one hour per week and from the very first week Joshua’s motivation escalated. His maths improved to the point where he is now often achieving 100% in his exams, he applies himself to do hours of homework, something he had rarely done in the past, but most of all Josh has found a love of mathematics. Maths is no longer a chore. I believe this all stems from Mr Henderson’s gifted ability not only as a teacher, also as an encourager and motivator. I get motivated just listening to them! Josh told us this week that he wishes he could be tutored every day after school. When the results come through from Joshua’s exams, he is delighted and this has inspired him to become motivated to spend much more time applying himself to his other subjects. Josh is working hard to gain a great result in his HSC. I can personally recommend Mr Henderson as a truly wonderful tutor.

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