Hans Walser, [20260706]

Sandpiles

Translated from German with AI

1 What it's about

Modeling erosion

Kinematics of the figures

2 Description

Iteration: We imagine several stacks of squares placed next to each other. Individual stacks may also be empty, meaning they have a height of zero. Furthermore, we choose an upper bound. If a stack is higher than this upper bound, we remove the top two squares and add one square each to the adjacent stacks on the left and right.

Interpreting the squares as grains of sand leads to a model of the erosion of a sandpile. Hence the title.

3 Examples

3.1 Different upper bounds

The first image (Fig. 1, far left) shows the initial situation. The upper bound 6 is indicated by the blue bar.

The following image shows the situation after the first step. The first stack, which was too high in the initial situation, has been reduced by two squares, which are now located as small stacks to the left and right of it.

The third stack, which was also too high in the initial situation, has also initially been reduced by two squares. One of these squares was given to the second stack of the starting situation, the other to the fourth stack. Since this stack was also too high, it had to transfer a square to the third stack, thus reducing the third stack by only one square overall. The fourth stack, which was also too high, initially had to give up two squares but received one from the third stack. This fourth stack was therefore reduced by only one square overall.

The number in the upper left of each figure indicates the current number of steps, and the number in the upper right indicates the total number of steps required to comply with the limit everywhere.

Fig. 1: Upper bound 6

Figure 2 shows the kinematics of the process at one-second intervals.

Fig. 2: Kinematics

If the upper bound is lowered to 5, the required number of steps also increases, coincidentally to 5 (Fig. 3).

Fig. 3: Upper bound 5

At upper bound 4, we need 9 steps (Fig. 4).

Fig. 4: Upper bound 4

At upper bound 3, we need 30 steps (Fig. 5).

Fig. 5: Upper bound 3

At upper bound 2, we need 61 steps (Fig. 6).

Fig. 6: Upper bound 2

At upper bound 1, we need 168 steps, which is almost three minutes (Fig. 7).

Fig. 7: Upper bound 1. Not one stone here will be left on another

For know-it-alls: How many steps are needed for upper bound zero?

3.2 Reducing Squares

Reducing a 4×4 square requires 35 steps (Fig. 8).

Fig. 8: Reducing a 4×4 Square

Reducing a 5×5 Square requires 95 steps (Fig. 9).

Fig. 9: Reducing a 5×5 square

3.3 Stepped gables

Fig. 10: Stepped gable

Fig. 11: Stepped gable

Weblinks

Hans Walser: Sandhaufen

https://walser-h-m.ch/hans/Miniaturen/S/Sandhaufen/Sandhaufen.html

Hans Walser: Sandpiles

https://walser-h-m.ch/hans/Miniaturen/S/Sandhaufen/Sandpiles.html