The full paper develops the necessary tools to analyse the interesting sequences in more detail and also summarises the motivating properties of Pascal's Triangle, including its fractal-like structure: colouring the odds and evens black and white results in the famous Sierpinski Triangle
, if zoomed out far enough.
As expected, the first resonance, mainly related to the external Sierpinski triangle
, remains unchanged for both antenna geometries, while the second one turns out to be more controlled by the inner and smaller Sierpinski triangle
, and consequently it is not correctly matched by the zero iteration fractal radiator.
Some are geometric and easy to spot, like a snowflake or the popular example known as the Sierpinski Triangle
, a triangle made of so many smaller triangles that the closer you look the more triangles you find.
So, in the Euclidean space dimensions, the Sierpinski Triangle
lies between the line (dimension 1) and surface (dimension 2).
Theoretically, the Sierpinski triangle is derived from an equilateral triangle ABC by excluding its middle (triangle A'B'C') and by recursively repeating this procedure for each of the resulting smaller triangles.
i], and therefore the area of the Sierpinski triangle is proportional to [lim.
Therefore the perimeter of the Sierpinski triangle is proportional to [lim.
For example, for the Sierpinski triangle (see Figure 1), its construction rule generates P = 3 miniature replicas, of scale 1 : 2 (i.
Counting activities suggested at the abstract level, for example, include counting the vertices, faces, and edges of polyhedra to infer a relationship; discovering the value of pi by measuring circumferences and diameters; and counting the number of triangles in successive stages of the Sierpinski triangle
The children individually created their own Sierpinski Triangle
(Figure 2) then joined them to make a very large Sierpinski triangle
By comparing the pattern of black cells (odd integers) to the shaded parts of the fractal called the Sierpinski triangle
, we were guided to the conjecture that as the number of rows of Pascal's triangle increases, so too does the percentage of even numbers, i.
In , the authors use Sierpinski triangles
to build a fractal Gasket-shaped frequency reconfigurable antenna.