Tessellating with Triangles

Consider a scalene right triangle (e.g. a 3-4-5). Two of them can be used to make a rectangle, which can then be used to tile an infinite plane.

In general, you can always put two triangles together as a parallelogram, which can then be used to tile an infinite plane. The reason is because adjacent angles in a parallelogram always add up to a straight line (180 degrees).

A lot of people are asking how shapes can tile an infinite plane. Here is my take: suppose that I give you a large plane. You tile the shapes to fill that large plane without leaving any gaps. Now, I scale the large plane to a bigger size. You still are able to fill that large plane by tiling the same shapes. If this is possible for a plane with any size, then that shape can tile an infinite plane.

We can use two of the triangles to form a parallelogram (just simply stick the same side together), and parallelogram can be used to tile an infinite plane

Construct a triangle where all three given points are midpoints of its sides.

Consider the tiling of a floor with identical equilateral triangles. Photographed perpendicular from above you see, well, identical equilateral triangles. Varying the angle of the camera you see the same floor, now with a perspectival distortion. By varying the angle you are able to create a tiling with identical scalene triangles of any kind.

The plane is infinite. It's ever expanding. So you can't have any shortage of space while putting any tiles and also there are no boundaries to the plane. So no matter what figure of finite dimensions you have, you can place copies of the figures in an infinite plane.

My take on this may seem too easy, but I nevertheless find nothing wrong with it. Consider any scalene triangle. By joining two such triangles using a common side you end up with an isosceles triangle. Using the fact that any infinite plane can be filled with isosceles triangles, which the author himself states, our isosceles triangle is enough to do the trick.

If you take 2 really got scalene triangles you end up with a rectangle and considering every one of a rectangles angles are equal, you could tile them into an infinite plane.

I am aware I am explaining the basic properties of rectangles, I just like to explain everything thoroughly.

It's true because two triangles will always form a parallelogram thus you can join them as an infinite plane.

Sticking two triangles together will make a quadrilateral. Any quadrilateral can tile up an infinite plane.

You can also always form a kite with two congruent scalene triangles, and kites can be used to tile an infinite plane.

Yes, always.

Because when you concatenate 2 scalene triangles on their hypotenuse you'll either end with one isoscele triangle or equilateral triangle.

The property therefore also applies to scalene triangles.

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you can always make parallelograms with two identical triangles. parallelograms will always tile since the opposite sides of a parallelogram are always parallel.

The way I thought of this was as follows: you can definitively prove that a figure can tile an infinite plane if that you can arrange the figure into a larger copy of itself. This can be accomplished with 4 copies of the scalene triangle by aligning the bottom edges of two triangles to form a single segment, then connecting the two top vertices with a third identical triangle, creating a 4th copy of the original shape, rotated 180 degrees, in the center. This creates a triangle similar to the one you started with, with twice the side lengths. Since it is identical in shape, this process can be repeated infinite times, covering an infinite plane.