# 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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As Jonas Meyer and others pointed out, you cannot expect to embed your points in the plane and preserve their distances. If you want to at least come close, there is a notion of distortion for an embedding of metric spaces, and algorithms for computing a minimum distortion embedding. I am no expert at this, so I won't attempt to suggest useful references, but searching those keywords may be helpful to you.

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If you want to keep the distances exactly, then no, not really. If you fix 2 points in the plane, and want a third point with prescribed distances from the first two, you only have two options for that third point (symmetric across the line connecting the first two points). This severely constrains even three points; more will almost surely be impossible to graph exactly.

If you want to alter the distances in a way to preserve an order on them (in which I mean: if the distance between a pair of points is greater than that of a second pair of points, this will remain the case after altercation), or see them in more than three dimensions but from various angles or perspectives, then there might be mathematics to help you out there.

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