Geometric series

How to use the applet

• Move the slider $N$ to display the first $N$ shaded triangles.
• Start with the large triangle $T_1$.
• In any triangle, connect the midpoints of its sides. The central (medial) triangle has side length scaled by $\tfrac12$, hence area scaled by $(\tfrac12)^2=\tfrac14$.
• In the applet, at step $k$ the shaded triangle has area

(relative to the area of the original triangle).

• As $N$ increases, you are adding areas

so the total shaded area approaches a limit.

• The construction partitions the current triangle into 4 congruent triangles each time: 1 shaded + 3 unshaded. Repeating this indefinitely shows the shaded region approaches one third of the whole triangle, matching the sum $\frac13$.

Description

Link to code.
This is quiet experimental and not related to the lecture notes. Paper reference: https://arxiv.org/pdf/2006.05062

This applet reproduces a classical picture that gives a visual proof of

Move the slider N to show the first $N$ shaded triangles. Each new shaded triangle represents the next term of the geometric series.

The construction uses the midpoint triangle: inside any triangle, connect the midpoints of the three sides. The small “central” triangle has side length scaled by $\tfrac12$, hence its area is scaled by $(\tfrac12)^2=\tfrac14$. In the applet, at each step we shade this midpoint triangle, then repeat the same construction in the smaller top triangle. Therefore the shaded areas are

and the total shaded area is their sum.

A second viewpoint explains why the total shaded area is $\tfrac13$ of the big triangle: at every step, the current triangle is partitioned into four congruent triangles. One of them is shaded, while the other three form the “remainder” to be subdivided at the next step. This creates layers whose total area is exactly three times the shaded area in each layer, so the union of all shaded triangles occupies one third of the whole triangle.

In the code, the midpoint construction is implemented by repeatedly computing midpoints and drawing the corresponding shaded triangle.

const TL = mid(T, L);
const TR = mid(T, R);
const LR = mid(L, R);
// shade triangle [TL, TR, LR] (area = 1/4 of [L, R, T])