## 3.6 Midpoint Displacement Method

The mathematical, non-linear and linear fractals presented above are deterministic, which means that repeating the transformations under the same starting conditions will always result in the same figures. The midpoint displacement method, however, belongs to the category of random fractals, such as the fractals generated by the DLA-method, which in general produce more nature-like "random" objects.

How does the Midpoint displacement method works? We draw a triangle on the screen and mark the centers of the three borderlines. Then we move these points perpendicularly to the lines up or down by a random factor. The resulting object consists of four smaller triangles, one of them is the combination of the three newly constructed points - see picture 20. The same is applied to the new triangles, and so on. For the decision whether the center point is moved up or down we use a coin. If the values of the displacement, the vertical intervals, decrease slowly, the resulting object will look like a "young" highly rugged mountain. In contrast to that, any fast decreasing of the displacement factor after only a few iterations produces a smooth mountain.

picture 20: Midpoint Displacement

The midpoint displacement method shows very clearly that the ideas around fractal geometry have already been known for a long period of time. Archimedes (287-212 B.C.) used the midpoint displacement for measuring the area under a parabola. For this purpose a vertical line is drawn from the center of the base line of the parabola until it touches the curve. This upper point is then connected with the base-points of the parabola. In the next step vertical lines are drawn from the centers of the two new outer lines until they again reach the curve. Once more they are closed to form triangles whose area can be given easily. Each step produces twice as many triangles as the step before. After some "iterations" the calculated area does not increase very much, that means that the remaining area between the triangles and the parabola becomes smaller and smaller. The height of a certain step is related to the heights of the steps before by the formula: height(n+1)=(1/4)*height(n).[ Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.74.]

picture 20: Midpoint Displacement

Moving up or down the midpoints of the sides of a triangle by a certain value can produce natural looking landscapes. The value of such a displacement can be calculated through different distribution rules - in this way rough or smooth looking mountain ranges can be generated. The construction of the final mountain-like surface requires an additional graphic procedure which connects all the generated points in the way that the space can be visualized - e.g. modeling by a rectangular network.

DAtTE - Detection of Attic Extensions: Workflow to analyze the potentials of roofs in an urban environment
Talk and Proceeding: eCAADe 2021 - Towards a new, configurable architecture (Novi Sad, Serbia, 2021 | hybrid conference)
presentation (video)

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