Figurate numbers: Difference between revisions

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-Most of us are familiar with triangular numbers, such as the arrangement of ten bowling pins in a triangle,
+#REDIRECT [[Activities/TurtleArt/Tutorials/Figurate_Numbers]]
-&nbsp;&nbsp;&nbsp;.<br>
-&nbsp;&nbsp;.&nbsp;.<br>
-&nbsp;.&nbsp;.&nbsp;.<br>
-.&nbsp;.&nbsp;.&nbsp;.
-or square numbers,
-.&nbsp; .&nbsp; .&nbsp; .<br>
-.&nbsp; .&nbsp; .&nbsp; .<br>
-.&nbsp; .&nbsp; .&nbsp; .<br>
-.&nbsp; .&nbsp; .&nbsp; .
-but we don't necessarily know how their arithmetic works, or whether we can do the same with, say, pentagons.
-Can we learn something by telling a turtle how to make these numbers? Let's try.
-==Linear numbers==
-Before we get to triangles, squares, and so on, we need to have a few tools, such as those provided in the [[Activities/TurtleArt/Tutorials/Counting|Counting]] tutorial, so that we can create figures and post notes about them. Then we can add a few more things.
-Two-sided polygons may or may not be allowed in a particular geometry, but they are not of interest here, because both sides are necessarily of the same length.
-[[File:TACounting.png]]
-==Triangles==
-What should we add? How about making dots of different colors, and providing notes about how many of each, plus the running total? Easy-peasy. We just need to do the arithmetic to work out where to put everything.
-[[File:TATriangular_Numbers.png]]
-The [[Activities/TurtleArt/Tutorials/Triangular_Numbers_Program|program to generate the triangular numbers]] takes up a lot of space, so I am putting it on another page.
-A problem that TA can't solve for us directly is the formula for triangular numbers. Gauss figured it out on his own as a child, but the rest of us need a hint, at least. Here is a big hint.
-[[File:TATriangularNumberFormulaHint.png]]
-The area of a rectangle can be written Base × Height. The area of a right triangle is half the area of a rectangle with the same Base and Height. Can you do the rest?