Figurate numbers - Revision history

Mokurai: New redirect

2011-07-14T05:31:14Z

New redirect

← Older revision		Revision as of 01:31, 14 July 2011
Line 1:		Line 1:
	~~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]]

	~~   .<br>~~
	~~  . .<br>~~
	~~ . . .<br>~~
	~~. . . .~~

	~~or square numbers,~~

	~~.  .  .  .<br>~~
	~~.  .  .  .<br>~~
	~~.  .  .  .<br>~~
	~~.  .  .  .~~

	but we don't necessarily know how their arithmetic works, or whether we can do the same with, say, pentagons. Pythogoras, the founder of Greek mathematics, is said to have discussed these figures and their relationships. We know that the triangle of 10 points, called the tetractys in Greek, was a central element in the Pythagorean religion.

	~~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.~~

	~~==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?~~

	~~==Other Figurate Numbers==~~

	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 in Euclidean geometry and Lobachevskian/hyperbolic geometry, and of indeterminate length in Riemannian/elliptic geometry, rather like cutting a great circle on a sphere into two pieces.

	~~[[File:TACounting.png]]~~

	~~We need separate pages for each of the remaining topics. Here you go.~~

	* [[Activities/TurtleArt/Tutorials/Square_Numbers\|Square Numbers]]
	* [[Activities/TurtleArt/Tutorials/\|Pentagonal Numbers]]
	* [[Activities/TurtleArt/Tutorials/Pyramidal_Numbers\|Pyramidal Numbers]]
	* [[Activities/TurtleArt/Tutorials/Cubic_Numbers\|Cubic Numbers]]

Mokurai: Add links, rearrange

2011-07-12T00:25:58Z

Add links, rearrange

@@ Line 13: / Line 13: @@
 .&nbsp; .&nbsp; .&nbsp; .
-but we don't necessarily know how their arithmetic works, or whether we can do the same with, say, pentagons.
+but we don't necessarily know how their arithmetic works, or whether we can do the same with, say, pentagons. Pythogoras, the founder of Greek mathematics, is said to have discussed these figures and their relationships. We know that the triangle of 10 points, called the tetractys in Greek, was a central element in the Pythagorean religion.
 Can we learn something by telling a turtle how to make these numbers? Let's try.
@@ Line 20: / Line 20: @@
 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.
 ==Triangles==
@@ Line 38: / Line 34: @@
 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?

Mokurai: /* Triangles */ Graphics. More to come.

2011-07-11T05:25:37Z

Triangles: Graphics. More to come.

← Older revision		Revision as of 01:25, 11 July 2011
Line 29:		Line 29:
	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.		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:~~TATriangularNumbers~~.png]]		[[File:TATriangular_Numbers.png]]

	[[~~File:TATriangularNumbersProgram.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.		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.

Mokurai: Add examples to stub

2011-07-10T22:58:24Z

Add examples to stub

← Older revision		Revision as of 18:58, 10 July 2011
Line 16:		Line 16:

	Can we learn something by telling a turtle how to make these numbers? Let's try.		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:TATriangularNumbers.png]]

			[[File:TATriangularNumbersProgram.png]]

			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?

Mokurai: Cleaner layout

2011-07-08T22:43:54Z

Cleaner layout

← Older revision		Revision as of 18:43, 8 July 2011
Line 1:		Line 1:
	Most of us are familiar with triangular numbers, such as the arrangement of ten bowling pins in a triangle,		Most of us are familiar with triangular numbers, such as the arrangement of ten bowling pins in a triangle,

	. . . .		.<br>
	. . .		. .<br>
	. .		. . .<br>
	.		. . . .

	or square numbers,		or square numbers,

	. . . .		.  .  .  .<br>
	. . . .		.  .  .  .<br>
	. . . .		.  .  .  .<br>
	. . . .		.  .  .  .

	but we don't necessarily know how their arithmetic works, or whether we can do the same with, say, pentagons.		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.		Can we learn something by telling a turtle how to make these numbers? Let's try.

Mokurai: New page stub

2011-07-08T22:30:12Z

New page stub

New page

Most of us are familiar with triangular numbers, such as the arrangement of ten bowling pins in a triangle,

. . . .
. . .
. .
.

or square numbers,

. . . .
. . . .
. . . .
. . . .

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.