Are there other ways to divide a square into equal parts (Equal as in matching.)

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Tatum Shearer

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Oct 20, 2022, 4:46:38 PM10/20/22
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I know you can split them by doing 4 lines horizontally or vertically, but I'm wondering if there are other ways to do this.

Barry Schwarz

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Oct 22, 2022, 9:42:26 AM10/22/22
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On Thu, 20 Oct 2022 13:46:37 -0700 (PDT), Tatum Shearer
<tshe...@bostonk12.org> wrote:

>I know you can split them by doing 4 lines horizontally or vertically, but I'm wondering if there are other ways to do this.

Any number of equally spaced horizontal or vertical lines will divide
the square into matching rectangles. Using both equally space
horizontal lines and equally spaced vertical lines will divide the
square into matching rectangles or squares. The tow diagonals will
divide the square into four matching triangles. There are also ways
to divide the square into matching jig-saw puzzle pieces.

What is your real question?

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John Francis

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Oct 22, 2022, 1:43:50 PM10/22/22
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In article <a6d91b3b-95e2-41f8...@googlegroups.com>,
Tatum Shearer <tshe...@bostonk12.org> wrote:
>I know you can split them by doing 4 lines horizontally or vertically, but I'm wondering if there are other ways to do this.

The most obvious other ways of doing this are to divide the square into four smaller squares (by drawing one horizontal and one vertical line), or into four triangles (by drawing two diagonals).

But that's just the start of what you can do! Both of those are special cases of the following general procedure:

o Draw a line from the centre (or center) of the square that intersects one edge, and projects some way beyond.
(The line doesn't need to be straight, but shouldn't intersect itself)

o Draw another copy of this line, rotated 90 degrees around the starting point.
(If this line intersects the first line you drew, give up and start over)

o Draw two more copies of the first line, rotated by 180 and 270 degrees.


This will divide the square into four equal pieces (and can be done in an infinite number of ways).


Note that this requires the smaller pieces to be rotated to make them all identical.
If rotations are not allowed then the first case - four smaller squares - still works.
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