Create the Sierpinski triangle on the transparency using the approach from Part I. At least three or four iterations are required. If you only draw one level, then the answer is no; there will be no filling.
The reason why you need at least three or four levels to fill in the image is because at each level of the triangle, some parts are removed, leaving holes that need to be filled in order to complete the image. So at least three or four levels must be drawn before the image can be said to be fully filled in.
The Sierpinski triangle has an infinite number of levels, but after some point it becomes difficult to create new levels because the triangles get smaller and smaller with each iteration.
This problem was solved by creating a hierarchy of triangles. At the lowest level of the hierarchy, which we'll call "level 0", there is just one triangle. This covers the entire surface of the transparency film. Each subsequent level is generated by taking the original level 0 triangle and dividing it into two equal parts. These nine new triangles make up level 1 of the hierarchy. They too cover the entire surface of the transparency film. The process is continued indefinitely so that every area of the transparency film is covered by a finite number of triangles.
The Sierpinski triangle exercise demonstrates core fractal principles: how a pattern may recur at different scales and how this complex shape can be generated by simple repetition. Each student creates his or her own fractal triangle, which is made up of smaller and smaller triangles. This game is best played in groups of between 3 and 10 students.
Students are invited to explore mathematics through play. Playing with numbers and shapes helps develop understanding of concepts that would not otherwise be exposed until later in a classroom session. The Sierpinski triangle activity is one example of math play that promotes creativity and problem-solving skills while encouraging young people to interact with their world around them.
Fractals were first described by Belgian mathematician Benoît Mandelbrot in 1977. He showed that many natural phenomena such as clouds, trees, coastlines, and even heart disease have similar patterns at each scale of magnification. The Sierpinski triangle exercise shows these hidden similarities by generating its own copies at each level of magnification.
This game requires no equipment other than pencil and paper. However, for an added challenge, you could extend the experience by having students create fractals in software such as Mathematica or GraphPad Prism. These programs allow players to change parameters such as scale and iteration count to see what effects this has on the output.
An isometric drawing is simple to create with a 30-60-90-degree triangle and a T-square, or with CAD programming. The lines that connect the corners of the square represent 90 degrees. The lines that connect opposite sides of the square represent 60 degrees. And the lines that connect the midpoints of opposite sides represent 30 degrees.
Isometric drawings are useful for showing three-dimensional objects from different angles without using complicated 3D techniques. For example, an artist could use an isometric drawing to show how a sculpture would look from all angles.
In mathematics, an isometry is a geometric figure whose corresponding points on its two surfaces (or more generally, between its n-dimensional spaces) are related by linear transformation; thus, isometries possess certain invariant properties. In the case of planar isometries, the term plane isometric refers to a map which preserves both distance and angle. Isometries play an important role in geometry, particularly in algebraic geometry, where they provide a connection between projective geometry and Euclidean geometry.
In physics, isometries are symmetries of physical systems that leave their relative positions unchanged. Thus, all atoms in a molecule remain fixed relative to each other when the structure is rotated around any central point.