Real life examples of ultrametrics or "the isosceles triangle principle"

Question

Mathematical Background and Definitions: The distinguishing feature of an ultrametric is the "strong triangle inequality" i.e. for all $x,y,z$,

$$d(x,y) \le \max(d(x,z), d(y,z)).$$

This implies that for any three points $x,y,z$, the two longer of the three pairwise distances $d(x,y), d(x,z), d(y,z)$ are equal, i.e. any triangle is isosceles. In yet other words, if $d(x,y) \neq d(y,z)$, then $d(x,z) = \max(d(x,y), d(y,z))$.

Relatedly, say there's a set $X$ with some bivariate "join" operation $(x,y) \mapsto x \odot y \in X$. (I deliberately use an abstract $\odot$ instead of the motivating example of an abelian group $(X,+)$, to allow for more examples.) I call an "ultrametric degree function" a function $\deg: X \rightarrow \mathbb R_{\ge 0}$ which has the property that $$\deg(x \odot y) \le \max(\deg(x), \deg(y))$$ and $$\deg(x) \neq \deg(y) \implies \deg(x \odot y) = \max(\deg(x), \deg(y))$$ but so that $$(*) \quad \text{ there are } x,y \text{ with } \deg(x \odot y) < \deg(x)=\deg(y).$$ (If it makes you feel better, switch $\le$ to $\ge$, $\max$ to $\min$ etc. and call such a thing a nonarchimedean filtration or something.)

Motivation and Question: I have studied $p$-adic numbers quite a bit, so I know how useful ultrametrics and nonarchimedean values are, mathematically. I do remember, though, how wildly counterintuitive the concept seemed at first. True, one gets used to it and develops an intuition after a while, but recently I have wondered if there are real life examples which convey the idea of an ultrametric, in particular the idea of the isosceles triangles principle. That is, they should give a feeling of "oh, that kind of 'distance' one can consider", to somebody whose math background does not go beyond a good high school education.

The reason I phrase my "ultrametric degree function" the way I do is that for this one, there is an example which, although mathematical, should be accessible to somebody with a good high school background: The usual degree function on polynomials (with $\odot = +$). A good high school student might see:

"OK, so the degree is a way to measure how "big" a polynomial is. Aha, yes, if I add a degree 7 and a degree 3 polynomial, the sum will have degree 7 again. Oh for sure, if I add any two polynomials, the result will just have the degree of the bigger one. [thinking pause] Oh wait, no, if I add $5x^2+x$ and $-5x^2-3$, then I get $x-3$, so the degree can get lowered. [thinking pause] Ah, but that can only happen if I start with two polynomials of the same degree, because the leading terms have to cancel out ... So, aha, if I add two polynomials of different degrees, then the bigger one decides, but if I add two of the same degree, then the result can get smaller."

I want examples of this principle from real life: "If I join two objects with different size, then the result will have the size of the bigger one; but if I join two objects with the same size, the result can become smaller." [Or the analogous with "bigger" and "smaller" reversed.]

Non-examples: Real life instances where the "bigger" (or smaller) always "decides" abound, because they come up when the "join" function just chooses the max (or min): tax class of a married couple (in some legislations); time since last accident in factory departments; highest score on attempts at a question. I was thinking in this direction at first, but the problem is that in all such cases, nothing interesting can happen if the two inputs have the same size; then the output will just have that same size too. So condition (*) seems the hard one to find outside of mathematics.

Examples: The Wikipedia article lists among "applications":

• [(1)] In condensed matter physics, the self-averaging overlap between spins in the SK Model of spin glasses exhibits an ultrametric structure, with the solution given by the full replica symmetry breaking procedure first outlined by Giorgio Parisi and coworkers. Ultrametricity also appears in the theory of aperiodic solids.
• [(2)] In taxonomy and phylogenetic tree construction, ultrametric distances are also utilized by the UPGMA and WPGMA methods.These algorithms require a constant-rate assumption and produce trees in which the distances from the root to every branch tip are equal. When DNA, RNA and protein data are analyzed, the ultrametricity assumption is called the molecular clock. [...]
• [(3)] In geography and landscape ecology, ultrametric distances have been applied to measure landscape complexity and to assess the extent to which one landscape function is more important than another.

I have read up a bit on (2) and like it very much; in fact, I will post an answer expanding that. I do not know anything about (1) or (3) and would be happy to learn more.

Answer 1

Here is a simplified version of how I understand example (2) from the question.

Consider all animal species which are to be found on earth today. We want some kind of measure of "how far one species is from another". We measure that distance by "how many million years ago did their last common ancestor live?".

For example, take Humans (H), Orangutans (O) and Common Newts (N). Say we get

$$d(H,O) = 17 , \quad d(H,N)= 320, \quad d(O,N)=320, $$

meaning that the last common ancestor of myself and any orangutan lived 17 million years ago (cf. https://en.wikipedia.org/wiki/Hominidae#/media/File:Hominoidea_lineage.svg), while to meet my last common ancestor with the newt in my garden pond I would have to travel back in time 320 million years. Note that from this alone, under current standard hypotheses of evolution, it follows already that the orangutan and the newt are also $320$ apart, because whatever common ancestor the newt and you have, that must also be a common ancestor for your partner and the orangutan; and conversely, that last common ancestor of humans and orangutans from 17 million years ago could, in principle, trace her lineage further back to the moment when our family split off from the proto-amphibians, whose grand-…-grand-child is our newt.

Thinking through this argument abstractly, we see that this "distance" between extant species indeed is an ultrametric. It is rather clear, in the example or in general, that $d(H,N)=d(O,N) = 320$ forces $d(H,O)$ to be bounded above by $320$, and allows it to be much smaller, indeed $d(H,O)=17$.

Apparently, biologists call the images which visualise such distances for a selecetion of species (images 1, 2) "ultrametric phylogenetic trees", cf. https://www.google.com/search?q=ultrametric+phylogenetic+tree .

From what I understand, ongoing research is "the other way around", i.e. they are trying to fit genetic data and fossil data, together with the molecular clock, to construct such trees, and "ultrametricity" is actually the goal: Meaning that, because we want the above common sense distance to work, there are some algorithms that produce reasonable estimates for how long ago what "branching" in the tree happened.

Please correct if something here is off.

Answer 2

We consider the public transport system of some city. The area is partitioned into zones $1,2,3,\ldots$, having the shape of concentric circles around the city center. For each zone $n$ there is a one-year ticket, covering all zones up to the $n$th one. The bigger $n$, the more expensive the ticket.

Not let $X$ be the set of all stations in the public transport, and define the distance $d(a,b)$ between two stations $a$ and $b$ as the price of the cheapest one-year ticket which allows you to go from $a$ to $b$. Then $d$ is an ultrametric on $X$ (assuming that all rides are done within the same year).

The ultrametric triangle equation is based on the property that, once a one-year ticket is bought to cover a certain connection, you can use it as often as you want for no additional costs.