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They used to meet in secret to discuss their mathematical discoveries.

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Only members were allowed to attend.

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They had a secret emblem, the pentagram.

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Let's see what the topic is for today.

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It was our old friend Pythagoras who discovered that the pentagram was full of mathemagic.

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The two shorter lines combined exactly equal the third.

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And this line shows the magic proportions of the famous golden section.

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The second and third lines exactly equal the fourth.

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Once again we have the golden section.

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But this is only the beginning.

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Hidden within the pentagram is a secret for creating a golden rectangle

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which the Greeks admired for its beautiful proportions and magic qualities.

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The star contains the golden rectangle many times over.

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It's a most remarkable shape.

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It can mathematically reproduce itself indefinitely.

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All these rectangles have exactly the same proportions.

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This figure also contains a magic spiral

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that repeats the proportions of the golden section into infinity.

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To the Greeks, the golden rectangle represented a mathematical law of beauty.

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We find it in their classical architecture.

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The Parthenon, perhaps one of the most famous of early Greek buildings,

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contains many golden rectangles.

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These same golden proportions are also found in their sculpture.

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These same golden proportions are also found in their sculpture.

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These same golden proportions are also found in their sculpture.

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In the centuries that followed, the golden rectangle dominated the idea of beauty and architecture throughout the Western world.

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The cathedral of Notre Dame is an outstanding example.

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The cathedral of Notre Dame is an outstanding example.

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The Renaissance painters knew this secret well.

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Think of a pentagram, Donald.

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Now put another inside, a third and a fourth.

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No pencil is sharp enough to draw as fine as you can think.

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Think.

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And no paper large enough to hold your imagination.

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In fact, it is only in the mind that we can conceive infinity.

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Mathematical thinking has opened the doors to the exciting adventures of science.

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I've never seen so many doors before.

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Each discovery leads to many others.

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An endless chain.

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What's the matter with these doors?

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Hey!

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These doors won't open.

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Get out!

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Of course they are locked.

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These are the doors of the future.

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And the key is...

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Mathematics!

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Right.

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Mathematics!

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The boundless treasures of science are locked behind those doors.

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In time, they will be opened by the curious and inquiring minds of future generations.