University of Tokyo 2002: Volume of Solids and Limits

Problem

In a coordinate space, take points P$${_k \left( \dfrac{k}{n}, \ 1-\dfrac{k}{n}, \ 0\right)}$$ , $${k=0,1, \cdots ,\ n}$$. Also, take points Q$${_k}$$ on the $${z}$$-axis where $${z \geqq 0 }$$ such that P$${_k}$$ Q$${_k=1}$$. Let V$${_k}$$ be the volume of the triangular pyramid OP$${_k}$$ P$${_{k+1}}$$ Q$${_k}$$. Find $${ \displaystyle {\lim _{n \rightarrow \infty}\sum _{k=0} ^{n-1}} }$$ V $${_k}$$.
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　The calculation should not be difficult. The coordinates of point Q$${_k}$$ can be found by calculating the distance, and the limit of the sum is in the form of a definite integral, resulting in $${\displaystyle {\lim _{n \rightarrow \infty} \sum _{k=0}^{n-1} V_k}=\dfrac{\sqrt{2}}{6} \int _0 ^1 \sqrt{x-x^2}dx }$$. The result is $${\dfrac{\sqrt{2}}{48}\pi}$$. Both can be found using standard methods.
　Now, what kind of solid is created? Of course, we cannot draw the figure when n is infinite, but it would be nice if we could draw it when n is a reasonably large number.
　That figure is the one in the heading. It is colored colorfully, so let's observe how it changes when n=1, 2, ...

Click the link to see the initial figure. This is for n=1.

At this time, P$${_0=(0, \ 1, \ 0)}$$, P$${_1=(1, \ 0, \ 0)}$$, and Q$${_0=(0, \ 0, \ 0)}$$, so no triangular pyramid is formed. Only the base is colored.

You can change the value of n with the slider on the right. Drag the green point to n=2 to see the following figure.

Since P$${_0=(0, \ 1, \ 0)}$$, P$${_1=(\dfrac{1}{2}, \ \dfrac{1}{2}, \ 0)}$$, P$${_2=(1, \ 0, \ 0)}$$, Q$${_0=(0, \ 0, \ 0)}$$, and Q$${_1=(\dfrac{\sqrt{2}}{2}, \ 0, \ 0)}$$, the triangular pyramid OP$${_1}$$ P$${_{2}}$$ Q$${_1}$$ is formed.

Let's try n=3.

Two triangular pyramids are formed, but since Q$${_1}$$ and Q$${_2}$$ are the same point (let's calculate it), they are triangular pyramids of the same height. The display also shows Q$${_1}$$ and Q$${_2}$$ overlapping.

Let's try n=4.

The situation changes. The height of the triangular pyramids changes. The light green and light purple triangular pyramids have the same height (Q$${_1}$$ and Q$${_3}$$ overlap), but the height Q$${_2}$$ of the light blue triangular pyramid is higher than that.

　As you increase the number of segments, it becomes a colorful figure. I think it is interesting just to observe this.

※ The figures are created with Cinderella (CindyJS).