Taxicab Geometry

A few weeks ago, I didn’t have my notebooks with me, so I started reading a book new to me, Taxicab Geometry. It was interesting, but I didn’t get very far into it. I tweeted about it. But, ended up getting frustrated by it.

During those tweets, I said it tied into product management, and that I’d blog about that. So this is that blog.

Non-Euclidean geometries take one or more rules or definitions and alters them. In Taxicab Geometry, consider yourself in some hypothetical, theoretical town that’s been laid out in a grid without any diagonal streets. Then, try to figure out the distance between two addresses are given that you can only go north, south, east, or west. In effect you have a digital geography.

In my earlier post, Building a Dog. Oh, Make that a Cat, I wrote about how to build an ontology measuring differences in bits, or measuring differences in terms of minimal marketable functionality. Both of these ways of laying out a geography tied into the notion of measuring the distance between a dog and a cat in an abstract sense. Calculating the distances would require a taxicab geometry.

In the dog and cat conceptual models, I claimed that there are sortables for concepts, but didn’t tell you what they were. So to find a more concrete example, I’ve looked into the word processing category to establish how MS Word and Notepad relate in taxonomic terms.

Taxonomies sort out realizations, things real or made as real as they can be. While ontologies sort out conceptualizations, things, or portions of things that have yet to be realized. Requirements capture conceptualizations or ontologies, while taxonomies classify delivered functionality. Taxonomies and ontologies coexist in an application. Taxonomies and ontologies are built up from classifying decisions, taxons and ontons for short. You end up with networks of these decisions, or decision trees, depending on the scope of the world you develop and compete within.

A Word Processor Taxonomy

A Word Processor Taxonomy for MS Word and Notepad

Our taxonomy of word processors, shows that the core functionality is the ability to edit text. Every word processor enables you to edit text, or it isn’t a word processor. A text box in any application provides rudimentary text editing functionality. This taxon sorts out the products competing in the word processing category from those outside the category.

We go on to further differentiate products, so additional taxons are needed to sort out the products within the word processing category. I wanted to sort out MS Word and Notepad, so the taxons were ordered to that end. The ability to search and replace, and move and copy functionality was all that was needed to complete the decision tree relative to Notepad.

Notepad does do page layouts, headers and footers, page numbers, stylesheets, or mail merge, but MS Word does. So each of these features becomes a taxon and helps us organize MS Word among its competitors. These taxons put distance between Notepad and MS Word.

This hierarchy demonstrates how software companies differentiate their products at the delivered functionality. But, software companies compete on ideas or concepts within conceptualizations, or at the ontology long before they compete on realizations. Technology adoption provides the nascent lifecycle for a conceptualization.

A word is turned to name the category, another to name the company, other words for new functionality. Our competitors come along and say yes, we are like them, but different. Our competitors define their ontologies differently. They add sortables. They define their lexicon slightly different as well. You end up locked in lexical warfare long before you sell your differentiated product. The differentiation changes and expands. The differentiation faces off not only in a one-to-one manner, but in a one-to-many manner as well as subtree and leaf fights, tree vs. tree.

Users construct their own conceptual models based on interactions with the realizations. They like this realization. A particular realization matches the conceptual model they had when they first started using the application better than another realization, another product. Better matches means incomplete matches, so these realizations have different costs relative to the use of the realizations. Eventually, if you application stays around long enough, an artifactual culture will spring up around you product’s ontology. Ontologies encode meaning. A product comes to mean something.

Even without considering our conceptual geographies we construct them. Taxicab geography is about distances within those geographies. So lets look at a map of an ontological geography and go for a drive calculating some distances as we go.

An Ontology on a Grid

An Ontology on a Grid

Here an ontology is represented by the black lines, branches, and squares. The root of the ontology is the black square at the top of the tree. Each sortable or onton is represented by the branching decision. The yellow and tan rectangles are measure of distance. Distance is measured up and down the tree (north/south) and across the siblings of each onton (east/west). Each black square is labeled by its distance from the root in black or gray. The light blue or orange numbers represent the distance associated with a measurement unit. The bottom number (left) in a measure represents the tree depth (north/south) distance. The top number (right) represents the sibling count. Both numbers are added together to arrive at the distance measure for a given concept, or black square.

The bottom right concept shows a distance of 8. It is on the forth row below the root, so its depth measures accumulated on the path from the root to itself is 4. In this representation, each of its ontons along the path assign it a sibling distance of 1, which in turn sums to 4 along the entire path. Both distances sum to 8.

The gray concept on the left, has a distance of 5. It’s distance from its immediate parent is one, with a sibling measure of zero. I was trying to stick with the gray grid as a means of measurement, but that didn’t work. I’ll redraw this figure with a strict unit measure. A zero sibling measure means that it is more specific than the immediate parent, the more general concept.

A zero sibling measure turns up in other ontons in this figure. They screamed at me that the representation wasn’t working. So I redrew the figure.

Ontology on a Grid with Unit Measure

Ontology on a Grid with Unit Measure

This figure differs from the previous figure in that a standard unit of measure is used everywhere, except where a single general concept is further defined by a single specific concept, the gray box of distance 5 (black). Further, the way the branching is represented gives a unique sibling distance to each sibling.

I also measured distances in different ways. The Stack (red) distance does not count siblings. The Sortables/Ontons (green) distance counts only the sortables/ontons (green outlines around relevant boxes) omitting the leaf. The Leaf (blue) counts includes siblings. Each of these different distances represent different geometries, typical of non-Euclidean geometries.

The book went into how circles would be represented. One of the exercises has you dividing up an city into school attendance zones. This was interesting, because it shows you how to allocate space in a category, aka market, or roles to competitors, complementors, and even open innovation. Moore pointed out in his technology adoption lifecycle books that a near monopolist is safe until they capture more than 74% of the market, which hints towards the 26% of the space that cannot be had. If you can’t have the space, you can still determine outcomes in that space.

I’ll get back to that book. The ontologies in the examples here would also be subject to the user’s cognitive limits, so the logarithmic schemes laid out in Cognitive models on the Efficiency Frontier, Innovation Visualization, and More on Innovation Visualization would apply, and would depend on how one counts, and the user’s knowledge of the underlying ontology.

Comments?

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3 Responses to “Taxicab Geometry”

  1. Requirements as Circles « Product Strategist Says:

    […] I’ve described built a geometry around bits in previous posts. See “Building a Dog. Oh, Make that a Cat”, “Now that you have that Cat”, and “Taxicab Geometry”. […]

  2. gulab Yadav Says:

    Very good answer

  3. A Spatiotemporal View | Product Strategist Says:

    […] so projections will underestimate the future. Hyperbolic geometries also give us things like a taxicab geometry with it’s trickier metric, which brings with it much risk, and the world lines of Einstein. […]

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