6 ms·

Colorized Math Equations

83 points by ghosthamlet 9y ago

aqsalose 9y ago
As I said when I spotted this on Twitter, I think highlighting multiple things at the same time is a bit confusing. Looking at the equations, I, for one, can't easily spot any connections between the colorized parts of the equation and the textual description; it becomes all the same rainbow porridge. What I'd like instead: highlight only a single thing in a single equation, and accompany the highlight with a text that explains it. Then repeat the (image with highlight, textual explanation) pairs until you've explained everything. Benefits: 1. I believe for many people like me, attention and focus on a single "highlighted" item at a time is much easier. 2. You can use only a single "highlight" color that can be colorbind friendly (it's enough that it is noticeably different than black) 3. You don't need to explain the equation in a single text passage (unlike in the examples, where it looks like the author wants to explain all of the equation in a single sentence so that all different concepts in the equation are represented in different colors both in a single text and the equation itself.) Also, in my experience, yellow/orange is terrible color for text / plots in print. (Peter Flach's machine learning textbook uses yellow extensively for similar "text highlights", and it made many sentences in that book frankly unreadable. I simply could not read light yellow text on white paper background.)
- dandare 9y ago
  It is very individual, the "rainbow porridge" works really for me, while your model may be better for other people. Finding distinct colors for multi-colored diagrams is a science on its own. They should have the same level of luminosity (so that they are readable on white background) and equal "distance" on the hue axis. The answer is the Munsell color scheme http://www.ktorides.com/wp-content/uploads/2015/01/colors-750x410.png http://www.ktorides.com/wp-content/uploads/2015/01/colors-75...
  - soegaard 9y ago
    https://en.wikipedia.org/wiki/Munsell_color_system https://en.wikipedia.org/wiki/Munsell_color_system
  - jhanschoo 9y ago
    If I use colors with uniform luminosity will color-blind have difficulty distinguishing them?
cm2187 9y ago
Mathematicians love greek letters, and if there are a couple of them it's easy enough to read the equation, but you can easily end up with 5 or more in a single equation, then have to spend extra brain time mapping which symbol refers to what variable. What I often do when I want to better understand a formula is to rewrite it by replacing greek letters with plain words. Less elegant but more readable.
- Terr_ 9y ago
  Yeah, people who aren't already familiar with the equation will be spending their precious and limited short-term memory on the mapping of symbols. By expanding the symbols to words, they have more mental resources to look at the relationships between terms. Essentially a lot of the same reasons not to use single-letter variables even in a statically typed language.
- TrinTragula 9y ago
  They don't "love" greek letters, you simply run out of letters. And you can't use words since that could be easily mistaken for a product between single variables. Once you learn them, things start getting way, way easier
  - cm2187 9y ago
    That would be the case if all the other letters in the alphabet were used in the equation. This is rarely the case. In fact more often than not, all letters of the greek alphabet are used.
    - aqsalose 9y ago
      By custom, Greek letters usually mean different things than letters of Latin alphabet. And also it's customary that certain letters in Latin alphabet are used for specific things. For example, Latin letter 'x' usually means a variable of interest / data point. Latin letter 'a' could be instead a constant or variable that is thought to be fixed for the time being. If you want more unknown variables in addition to 'x', you can use y, z, maybe w, but if you want more than that you are out of letters and are often better off with super/subscripting (if it makes sense in the context), because a, b, c, d would be interpreted as constants. Especially in statistics/ML x is reserved as symbol for the independent variable and y is reserved as symbol for the dependent variable; z, w are common for latent variables. f,g,h are common shorthands for functions. i,j are double-booked as an "arbitrary index symbol" or imaginary units. k,m,n are also constants, and often reserved specifically as constants that denote the dimensionality or sequence lengths (m times n matrix, sums x_1 + ... + x_n, sequences (x_1, ..., x_n), etc). (k can be also be an index in addition to i and j.) o is easily confused with 0 or a circle that is usually a function composition sign (I usually see the letter o only in the CS small-o-notation context, rendered in fancy typeface). p,q,k are again constants or sometimes alternative indices in complicated sums where i and j are not enough, but from p to v you have relatively underused letters. However, they are not totally "free" either: s and t and r and l are again common symbols for scalar variables like length or time or rotation; in statistics, p often is reserved for probabilities (conditional probability of x given m is p(x|m) etc). In linear algebra u and v can be arbitrary vectors, or unit vectors, or in complex analysis they are real / imaginary parts of a complex valued function. Upper case letters are reserved for matrices and operations or (especially consonants like K or C) sometimes again constants. In other fields things might go differently, but the general gist is that I do often feel like the alphabet is not enough. Most often Greek letters get quite much use when you need symbols for angles or rotations (for example, Fourier transformation pairs) or as (hyper)parameters of probability distributions.
catnaroek 9y ago
This approach eventually breaks down. Could you give a useful colorized explanation for Urysohn's metrization theorem? Or the Vitali covering lemma? Colors are useless. Just learn to think abstractly. This is what (supposedly) distinguishes you from lesser animals.
- threatofrain 9y ago
  Even if this is true, the way you deliver insults probably sets back the message.
  - catnaroek 9y ago
    I'm not arguing against examples. I'm arguing against using colors to make up for the inability to understand abstract structures. In my experience, the people who argue for this are usually “math educators” who don't do much mathematics themselves.
- gugagore 9y ago
  I hope most people care about understanding something more than they do distinguishing themselves in your view. For many people, concrete examples can help them understand an abstract idea. Even if someone doesn't grasp the full abstract idea, at least they might have a concrete example that exemplifies the idea. I'm not familiar with your examples, but it is true that complex nuggets of math have evolved to be simpler, more elegant and sometimes more powerful. I try not to pride myself to much for understanding something complicated and abstract, because it would be even better if I had a way to conceptualize of it that made it simple and obvious. Finally, most _things_ breakdown. Doesn't mean that they aren't useful in a limited context.
  - catnaroek 9y ago
    > For many people, concrete examples can help them understand an abstract idea. Sure enough. > Even if someone doesn't grasp the full abstract idea, at least they might have a concrete example that exemplifies the idea. Examples are only useful inasmuch as they pave the road to understanding abstract definitions. On their own, examples are pretty much useless. In any case, your comment has nothing to do with the original topic, namely, whether coloring different parts of mathematical definitions is particularly helpul. It seems to me that the real problem here is that a certain kind of people has great difficulty understanding the syntactic and semantic structure of mathematical definitions. Now: (0) The syntactic structure of mathematical definitions is the usual syntactic structure of English, or whatever your first language is. If it hasn't been drilled into your head in elementary school, sue your elementary school. (1) As for the semantic structure of mathematical definitions, well, I guess there is no way around thinking hard about it, annoying as though it may be. > but it is true that complex nuggets of math have evolved to be simpler, more elegant and sometimes more powerful. I try not to pride myself to much for understanding something complicated and abstract Simplicity usually comes from having better abstractions (e.g., how Stokes' theorem evolved to its modern version in terms of differential forms), not from rejecting abstractions altogether.

83 points by ghosthamlet 9y ago

As I said when I spotted this on Twitter, I think highlighting multiple things at the same time is a bit confusing. Looking at the equations, I, for one, can't easily spot any connections between the colorized parts of the equation and the textual description; it becomes all the same rainbow porridge. What I'd like instead: highlight only a single thing in a single equation, and accompany the highlight with a text that explains it. Then repeat the (image with highlight, textual explanation) pairs until you've explained everything. Benefits: 1. I believe for many people like me, attention and focus on a single "highlighted" item at a time is much easier. 2. You can use only a single "highlight" color that can be colorbind friendly (it's enough that it is noticeably different than black) 3. You don't need to explain the equation in a single text passage (unlike in the examples, where it looks like the author wants to explain all of the equation in a single sentence so that all different concepts in the equation are represented in different colors both in a single text and the equation itself.) Also, in my experience, yellow/orange is terrible color for text / plots in print. (Peter Flach's machine learning textbook uses yellow extensively for similar "text highlights", and it made many sentences in that book frankly unreadable. I simply could not read light yellow text on white paper background.)

dandare 9y ago
It is very individual, the "rainbow porridge" works really for me, while your model may be better for other people. Finding distinct colors for multi-colored diagrams is a science on its own. They should have the same level of luminosity (so that they are readable on white background) and equal "distance" on the hue axis. The answer is the Munsell color scheme http://www.ktorides.com/wp-content/uploads/2015/01/colors-750x410.png http://www.ktorides.com/wp-content/uploads/2015/01/colors-75...
- soegaard 9y ago
  https://en.wikipedia.org/wiki/Munsell_color_system https://en.wikipedia.org/wiki/Munsell_color_system
- jhanschoo 9y ago
  If I use colors with uniform luminosity will color-blind have difficulty distinguishing them?

cm2187 9y ago

Mathematicians love greek letters, and if there are a couple of them it's easy enough to read the equation, but you can easily end up with 5 or more in a single equation, then have to spend extra brain time mapping which symbol refers to what variable. What I often do when I want to better understand a formula is to rewrite it by replacing greek letters with plain words. Less elegant but more readable.

Terr_ 9y ago
Yeah, people who aren't already familiar with the equation will be spending their precious and limited short-term memory on the mapping of symbols. By expanding the symbols to words, they have more mental resources to look at the relationships between terms. Essentially a lot of the same reasons not to use single-letter variables even in a statically typed language.
TrinTragula 9y ago
They don't "love" greek letters, you simply run out of letters. And you can't use words since that could be easily mistaken for a product between single variables. Once you learn them, things start getting way, way easier
- cm2187 9y ago
  That would be the case if all the other letters in the alphabet were used in the equation. This is rarely the case. In fact more often than not, all letters of the greek alphabet are used.
  - aqsalose 9y ago
    By custom, Greek letters usually mean different things than letters of Latin alphabet. And also it's customary that certain letters in Latin alphabet are used for specific things. For example, Latin letter 'x' usually means a variable of interest / data point. Latin letter 'a' could be instead a constant or variable that is thought to be fixed for the time being. If you want more unknown variables in addition to 'x', you can use y, z, maybe w, but if you want more than that you are out of letters and are often better off with super/subscripting (if it makes sense in the context), because a, b, c, d would be interpreted as constants. Especially in statistics/ML x is reserved as symbol for the independent variable and y is reserved as symbol for the dependent variable; z, w are common for latent variables. f,g,h are common shorthands for functions. i,j are double-booked as an "arbitrary index symbol" or imaginary units. k,m,n are also constants, and often reserved specifically as constants that denote the dimensionality or sequence lengths (m times n matrix, sums x_1 + ... + x_n, sequences (x_1, ..., x_n), etc). (k can be also be an index in addition to i and j.) o is easily confused with 0 or a circle that is usually a function composition sign (I usually see the letter o only in the CS small-o-notation context, rendered in fancy typeface). p,q,k are again constants or sometimes alternative indices in complicated sums where i and j are not enough, but from p to v you have relatively underused letters. However, they are not totally "free" either: s and t and r and l are again common symbols for scalar variables like length or time or rotation; in statistics, p often is reserved for probabilities (conditional probability of x given m is p(x|m) etc). In linear algebra u and v can be arbitrary vectors, or unit vectors, or in complex analysis they are real / imaginary parts of a complex valued function. Upper case letters are reserved for matrices and operations or (especially consonants like K or C) sometimes again constants. In other fields things might go differently, but the general gist is that I do often feel like the alphabet is not enough. Most often Greek letters get quite much use when you need symbols for angles or rotations (for example, Fourier transformation pairs) or as (hyper)parameters of probability distributions.

catnaroek 9y ago

This approach eventually breaks down. Could you give a useful colorized explanation for Urysohn's metrization theorem? Or the Vitali covering lemma? Colors are useless. Just learn to think abstractly. This is what (supposedly) distinguishes you from lesser animals.

threatofrain 9y ago
Even if this is true, the way you deliver insults probably sets back the message.
- catnaroek 9y ago
  I'm not arguing against examples. I'm arguing against using colors to make up for the inability to understand abstract structures. In my experience, the people who argue for this are usually “math educators” who don't do much mathematics themselves.
gugagore 9y ago
I hope most people care about understanding something more than they do distinguishing themselves in your view. For many people, concrete examples can help them understand an abstract idea. Even if someone doesn't grasp the full abstract idea, at least they might have a concrete example that exemplifies the idea. I'm not familiar with your examples, but it is true that complex nuggets of math have evolved to be simpler, more elegant and sometimes more powerful. I try not to pride myself to much for understanding something complicated and abstract, because it would be even better if I had a way to conceptualize of it that made it simple and obvious. Finally, most _things_ breakdown. Doesn't mean that they aren't useful in a limited context.
- catnaroek 9y ago
  > For many people, concrete examples can help them understand an abstract idea. Sure enough. > Even if someone doesn't grasp the full abstract idea, at least they might have a concrete example that exemplifies the idea. Examples are only useful inasmuch as they pave the road to understanding abstract definitions. On their own, examples are pretty much useless. In any case, your comment has nothing to do with the original topic, namely, whether coloring different parts of mathematical definitions is particularly helpul. It seems to me that the real problem here is that a certain kind of people has great difficulty understanding the syntactic and semantic structure of mathematical definitions. Now: (0) The syntactic structure of mathematical definitions is the usual syntactic structure of English, or whatever your first language is. If it hasn't been drilled into your head in elementary school, sue your elementary school. (1) As for the semantic structure of mathematical definitions, well, I guess there is no way around thinking hard about it, annoying as though it may be. > but it is true that complex nuggets of math have evolved to be simpler, more elegant and sometimes more powerful. I try not to pride myself to much for understanding something complicated and abstract Simplicity usually comes from having better abstractions (e.g., how Stokes' theorem evolved to its modern version in terms of differential forms), not from rejecting abstractions altogether.