The Truth About Infinitycjbalm/Quest/Breakouts/Infinity.pdf · The Truth About Infinity.pptx...
Transcript of The Truth About Infinitycjbalm/Quest/Breakouts/Infinity.pdf · The Truth About Infinity.pptx...
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The Truth About In-inity QUEST Summer 2014 Jenny Johnson, [email protected]
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“The infinite! No other quesBon has ever moved so profoundly the spirit of the man; no other idea has so fruiIully sBmulated his intellect; yet no other concept stands in greater need of clarificaBon than that of the infinite.” -‐ David Hilbert
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In-inity seems to come up not just in mathematics, but in science, art, and religion.
M. C. Escher
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Mathematics
• EsBmaBng the circumference of a circle by inscribing polygons.
• Build up to conBnuously compounding interest.
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Fractals
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Sierpinski Triangle The Sierpinski Triangle is made by repeatedly removing the middle triangle from the prior generaBon. The number of colored triangles increases by a factor of 3 at each step.
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Koch Curve The Koch curve is made by repeatedly replacing each segment of a generator shape with a smaller copy of the generator. At each step, the total length of the curve gets longer, eventually approaching infinity.
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The Koch Snow-lake
A curve of infinite length that encloses a finite area.
Draw a circle around your snowflake, will the snowflake ever grow outside your circle?
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Geometric Series Sum infinitely many posiBve numbers:
What is the sum of these infinitely many posiBve numbers? Use a number line to show that the sum is finite.