The Klein Bottle Promise
Explanation of the significance of this Substack's logo ... and an offer to receive a Klein bottle
Geometrical structures can be paradoxical. This Substack’s logo shows a schematic representation of a Klein bottle. That is a geometric surface, for which it is possible to move from a point on the inside, along the surface, and back to the same point on the outside. More about that later. For now, suffice it to say that a Klein bottle would be most accurately represented in four dimensions. Some of the strangest geometric behaviors can be found in spaces with more dimensions than the familiar three spatial ones. Not only mathematicians are interested in such geometries. Data scientists routinely work with what they call high-dimensional data. The main purpose of this post is to discuss the paradoxical structure of people’s preferences as a problem of geometry. Also to give away Klein bottles ...
Even three-dimensional spaces can be paradoxical. A mathematically exact straight line is infinitely long. The universe is almost certainly not infinite in the sense of accommodating such a line. The same is true for a mathematically exact flat plane. Even the most basic geometrical structures exceed the entirety of the physical space known to us. Fortunately, Planet Earth is not a plane. The beauty of spheres is that they have finite surfaces that can be traversed many times without ever reaching an end. Earth’s surface is not only finite but constant. It does not expand. The universe expands, and initially the expansion was exponential. Something that grows by a constant percentage in each step shows exponential growth. Many quantities on Earth expand exponentially: The amount of information, the power of computers, the total monetary value. Earth’s surface is constant. How monetary value can grow exponentially while Earth’s surface is constant is a story in itself. The following observations focus on geometry.
The geometry of some types of information is tied to physical space. Cities, states, countries, and continents break up information spaces just as they break up physical space. Two people living in the same city are mostly subject to the same laws. However, location does not determine everything. Chess players can live anywhere in the world. So can people who don’t care about the game of chess. Almost anywhere, there are some people who appreciate embroidery. And people who do not like embroidery at all. One could graph people based on how much they enjoy chess and to what extent they appreciate embroidery. A graph of likes and dislikes is a bit like a map. A physical map graphs landmarks or people by their latitude and longitude in physical space. Replacing latitude and longitude with interests allows creating maps of preferences.
Any person could be characterized by hundreds if not thousands of likes and dislikes. Unfortunately, we cannot physically draw a map that represents thousands of dimensions at the same time. Conventional maps are two-dimensional. In the space of preferences, a two-dimensional map represents no more than two independent likes or dislikes. Graphing even three would require more than a flat map. A virtual reality headset would work for the specific problem of mapping three preferences. Fancy visualization tools, as they can be seen in some movies, would too. It would be stunning to walk within a three-dimensional visualization of information. It would also fail to, by itself, solve the problem of visualizing dozens or more information dimensions.
Data scientists routinely work with thousands of dimensions of data. They treat columns in databases as dimensions, and databases can have many columns. In some ways, information spaces are quite similar to physical spaces. In Manhattan, addresses allow estimating distances. The differences in street and avenue numbers add up to the number of blocks that it takes to get to a destination. In data science, the number of mismatches in preferences provide some measure of how differently another person thinks. The parallels between spatial and information dimensions go further than distance calculations. In a city or town, one can estimate the number of neighbors who live within one block, two blocks, or three blocks of one's location. Doing so is equally possible in physical and in information spaces.
This is where things get weird. Mathematically, high-dimensional information spaces are so much like low-dimensional physical ones that we may get duped into thinking we understand them already. It’s not quite so easy. In a city, the number of neighbors, one, two, or three blocks away increases gradually. In a thousand-dimensional information space, mostly, there are no immediate neighbors. There are not even close neighbors who are, say, within three or four blocks. Rather, everyone suddenly becomes a neighbor all at once. The increase in neighbors with growing radius is so dramatic that it appears like a step. For one radius almost no-one is “in.” Increase the radius just barely, and almost everyone is. That makes it difficult to identify groups. Two people who are somewhat close to a third person may not be close to each other.
For those of us who live in cities, this may not be surprising. In urban areas, it is common to make friends based on interests. People know that place of origin does not determine much. Someone from a far away place may share many interests. Someone from the same hometown may share fewer. It is extremely unlikely that two people agree on all their preferences. Just because I consider two people to be friends, does not have to say much about those two friends' mutual interests. Any two people are expected to share some but not all preferences.
In rural areas the situation can be different. Small-town gossip is strongly tied to physical space. A person who lives further away is also more distant in the space of conversation topics. Those who go to the same church share many values. Even if we were to consider each value as a different dimension, those dimensions may not be independent. For those who let all their preferences be shaped by the same group, any number of dimensions may reduce to one or very few. The resulting distribution of preferences may appear quite like the spatial cluster of places in a town with its inherently two-dimensional map. Anyone’s ostensible alignment with group values can be quantified like the distance to the town center. Those who are within the boundaries are in. Anyone else is out. Someone who grew up in a different country, possibly going to a different church, eating different food, and wearing different clothes is bound to be considered different.
High-dimensional information spaces are fascinating and hard to understand. “In” and “out” lose their meaning when the number of neighbors may increase almost like a step. The difficulty is not only that two people may be inside of my circle but each may be outside the other’s. The space of interests may also be shaped in ways that are hard to picture within the three spatial dimensions. Imagine breaking up an information space based on which side of a dividing line a person is on. Now imagine that the dividing line takes the shape of a Möbius strip. You can construct a Möbius strip by cutting a wide rubber band and reattaching the edges after rotating one of them sideways by 180 degrees. Run your finger along the strip and you will reach the point at which you started but on the opposite side. For any two points on your Möbius strip you can argue that they are on the same side of the rubber band.
Two Möbius strips can be joined to create a shape that looks like a bottle. A bottle for which the spout passes through the body on one side and then merges with it on the other. This structure is called a Klein bottle after the mathematician Felix Klein, who first described it in 1882. An object that touches the inside of a Klein bottle may be moved along the surface until it touches the outside in the same location in which it started. It shares this property with the Möbius strips from which it is constructed. A mathematician may point out that, strictly speaking, a closed, non-orientable, and non-intersecting surface requires four dimensions for an accurate representation. The Klein bottle can be approximated well enough in three dimensions to allow building a model. Poetically, it can be viewed as a messenger from high-dimensional spaces, communicating their complexity.
High-dimensional information spaces allow for surfaces much more complex than Möbius strips or Klein bottles. Distinguishing “in” and “out” is often not meaningful. Reasoning within such spaces is harder than discussing problems that can be mapped into two dimensions. It is easy to make mistakes when dealing with complicated geometries, physical or otherwise. That includes this writing. What I write may contain mistakes, whether it is about geometry or not. Please point me to anything you consider wrong! As a thank you, I will send you a knitted, sewed, or otherwise constructed Klein bottle for any document on which you comment. Do I have enough styles to send you a different one each time? We shall see.
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