# Given a metric function between a set of abstract points, what is the best way to plot them on a 2D space?

I have a list of several entities, all of each have a numerical relationship to each other that defines an abstract distance.

Is there a mathematical way to plot all of these on a 2D space, turning the abstract distance into a 2D euclidian distance, and preserving all of their distances?

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You might be asking whether every finite metric space can be isometrically embedded in the plane. According to Wikipedia (with 2 references given), it is not even possible in general to isometrically embed a finite metric space into Eucidean space of any dimension: en.wikipedia.org/wiki/Metric_space#Examples_of_metric_spaces –  Jonas Meyer Dec 13 '10 at 7:58

The question really asks if there is an isometry of the metric space into the plane. We will give a counter-example to show that this cannot always be done.

Consider a metric space with $4$ points in which the distance between any $2$ distinct points is $1$. After placing $2$ points in the plane, the third must lie in the intersection of the circles of radius $1$ centred at these $2$ points. The fourth must lie in the intersection of the circles of radius $1$ centred at the other $3$ points, and there is no way to place it.

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If your purpose is visualization, then often a good approximate solution is useful. This is called Multidimensional Scaling (MDS).

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