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## Draw and Turn...Repeating...

Start with several sheets of graph paper and a repeating decimal like 0.123123...

At grid intersection, mark your starting point with a big dot.

Working from the repeating decimal...

• Draw a segment up one unit (because 1 was first digit)
• Turn right 90o
• Draw a segment straight (i.e. right) two units (because 2 was second digit)
• Turn right 90o
• Draw a segment straight (i.e. down) three units (because 3 was third digit)
• Turn right 90o
• Draw a segment straight (i.e. left) one unit (because 1 is first digit of next repeating bllock)
• Continue process...rotating and drawing lengths according to the repeating decimal 0.123123...
What pattern do you see? Would you get the "same" pattern with the repeating decimal 0.231231...?

Build patterns for these numbers: (a) 0.2222... (b) 0.121212... (c) 0.12341234... (d) 0.1234512345... (e) 0.102102... (For digit 0, make a 90o turn, but move no distance)...What do you notice?

Try 7ths: 1/7 = 0.142857...; 2/7 = 0.285714...; etc. What geometric principles are in play?

What does it take for a repeating decimal to have a closed path--one where the repeated digits will repeat a particular path?

What would you expect to happen with the digit patterns for the numbers π or e? Why?

Could a repeating decimal create a line segment that keeps increasing in discrete jumps?

What decimal would create the diagram shown above?

Now, make up your own questions for exploration....

Note: These explorations are part of the idea of Spirolaterals.

Source: From a conference workshop given by Bob Kansky (WY), a good and respected friend.

Hint: No hint needed...just start exploring....

Solution Commentary: No solution commentary needed...as the patterns and exploration guidelines are in front of you.