# How good is your geometric visualization?

Mine is not all that good. There is something I just realized and I am still having trouble seeing why it works.

Suppose you have an equilateral triangle that you want to divide into 9 smaller triangles, 1 on top, 3 in the next row, and 5 on the bottom row. Is it obvious what the simplest way of doing this is? It was not for me, and it only occurred to me after thinking about it.

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## 9 Answers

I didn’t really have to think about it. As soon as I read your question I knew how the subdivision must work and I could see in my mind’s eye the top two layers fairly clearly and the bottom row more vaguely. I have seen pictures of the Sierpinski triangle which may be what my imagination used.

Okay, but how would you draw it? Given a drawing of the triangle, do you see how it could be subdivided by drawing only 6 additional lines.

No, there would need to be 8 lines the way I picture it.

It only takes 6 lines. Draw the 2 lines that form the bottom of the first and second rows. Rotate the triangle by 120 degrees and draw the bottoms of the first and second rows relative to the new base. Rotate another 120 degrees and repeat. The 6 lines are all that are needed to form the triangles. This is not at all obvious to me, but I made a drawing and it works.

Here are some other interesting aspects of this relatively simple drawing. The sides of the small triangles are ⅓ the sides of the larger triangle. That means their area is 1/9 of the total area, which means there must be 9 small triangles. The numbers of triangles in each row are consecutive odd numbers, which always add up to a perfect square. We can generalize the procedure to divide each side of the triangle into n parts using 3(n-1) lines, creating n**2 triangles.

I see it after drawing it but you have to assume that lines are contiguous at the intersections so it’s a trick question.

It is analogous to dividing a square. To get 9 subsquares, draw 3 horizontal and 3 vertical lines.

Well, I think it would be obvious if I had paper and pen, or blocks.

As it was, I did have a visual that came up as you were describing it, because I remember seeing such arrangements before.

And, I look Logical Geometry in 6th grade, was pretty good at it, and have a good memory.

But it wasn’t *immediately* obvious, and for a problem of that type, I would either want to sketch it or use blocks if I were to be certain my mental image was accurate, and/or to take time to carefully imagine it and reason about it. But I can reason it out in my head without a visual aid, but also having done geometry before and remembering a fair amount of it helps a lot.

For instance, I know and can easily visualize and reason from properties and theorems I remember about equilateral triangles, and from experience with them on paper and in blocks, etc., that the top tip triangle would share a side with an equal triangle, and then if you added two more such triangles, each sharing one of the other two sides of that second triangle, that together they’d be four triangles that make up one larger triangle with sides twice as long as the component triangles. Then you’d just extend it, with the two triangles with sides on one side sharing sides of two new triangles, another triangle between those, and then two triangles at the tips, for a triangle made of nice triangles with side length three times that of the component triangles.

Easy for me. 6 lines.

Edit: although, recently on another Q about counting triangles I screwed it up and had the wrong answer, so maybe you got me on a good today. Lol.

Two final points.

The method works if the original triangle is not equilateral.

If you give each parallel set of lines a different color, each triangle will have one side of each color.

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