A color wheel or five-dimensional color space?

The most interesting thing about the card game Magic The Gathering is the color wheel. I do not know of a game or even a book or movie series in fantasy that is rooted so strongly in clear and strong archetypes that have such rich underlying philosophies.

Magic has five colors and each of these has its own niche of card abilities, game effects but also its own flavor. For instance, blue creatures can fly but green cannot. In an extreme and schematic summary:

  • Black cares about power and accumulates power by doing things the other colors consider taboo such as dealing with death,disease and deceit. Magic designer Mark Rosewater discussed black further here.
  • Blue cares about self-perfection and reaches this by knowledge, manipulation of others and artifice. It is also associated with the elements of water and air. Rosewater talks about blue further here.
  • White cares about peace and seeks to establish this through order, whether military, religious, moral or legal. It is also associated with the light. Rosewater reflects on white here.
  • Green cares about harmony and believes that natural growth, accepting reality and spirituality lead to this. Rosewater talks about green here.
  • Red cares about freedom and therefore embraces emotions, violence and chaos. It is associated with the elements of earth and air. Rosewater discusses red here.

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The color wheel is often represented as it is beside here: each colour has two allied colors (next to it) and two opposite colors (on the other side). Green is allied with white and red, but is an enemy of blue and black. The question that I seek to answer here, is whether the differences between these colors are indeed structured in this fashion. Is it simply the case that every colour has its own niche abilities; or is it the case that there is an underlying structure of similarities and differences that the allied colors have more overlap than the enemy colour pairs. The question is whether in terms of their overlapping abilities there is ‘color wheel’-like structure. In this color wheel:

  • White is more similar to blue and green than it is to red and black. For instance, white and green are more committed the community than the other colors.
  • Blue is more similar to white and black than it is to red and green. For instance blue and green differ strongly in the way they approach artificial things (green hates them but blue embraces them).
  • Black is more similar to blue and red than it is to white and green. Black and red are more inclined to kill living things than the other colors.
  • Red is more similar to black and green than it is to white and blue. Red and white differ on the dimension of chaos versus law.
  • Green is more similar to red and white, that it is to black and blue. Green and red both have a natural ferocity to them.

To me the identities of ‘red’ and ‘white’ are as clear as terms as ‘left’ and ‘right’ in politics. These are actually very similar. For instance, they are not well-defined: every gamer/citizen has an intuitive understanding of them but these may be slightly different. Moreover, their meanings can shift: cards that used to be clearly blue in Alpha such as Prodigal Sorcerer, are now, more than twenty years later clearly in another color (red in this case). Historically, the left in the Netherlands championed the separation of church and state, now more and more the right emphasizes this issue. Moreover, interesting things can happen in the overlap between colors that is when magic designers make two-color or hybrid cards or when politicians have to form coalitions.

In order to analyse the color wheel we look at keywords. We limit our analysis to keywords that have been used in more than one block of Magic (excluding the Time Spiral Block). We thus include evergreen keywords (‘flying’), recurring keywords (‘cycling’) and phased-out keywords (‘banding’). We count the number of cards that has each keyword per colour. For instance, there are thirteen blue cards that mention ‘first strike’ and 143 white cards that mention ‘first strike’. We do this for all cards in gatherer.

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Is this a good way to measure the extent to which colors are different and similar? It may be that a lot of the green mentions of “flying” come from flying-hate cards or that the blue mentions of first strike are from off-color activations. To establish the validity of our measure we correlate the number of cards with keywords per color with the extent to which Mark Rosewater has identified colors as being primary, second, tertiary etc. in a recent “Drive to Work“. As can be seen in the figure next to this texts, these are strong negative correlations between the number of cards with an ability and these judgements by Rosewater: that is cards that are primary in an ability tend very strongly to have most cards with this ability. The only exception is menace. Here the correlation is still -0.43, but the colour that is primary in this keyword according to Rosewater (black) has only three cards with this ability compared to red. But this may be the effect of the fact that this ability is relatively recent.

The data that we have, is a cross-table: the number of cards with a particular keywords in particular color. To a political scientist, this data calls out “correspondence analysis“. A method specifically developed to visualize this kind of data, to assess its dimensionality and to look at differences and similarities between the categories in the analysis.

Rplot1 copy The first step is assessing the ‘dimensionality’ of the data. That is understanding how many dimensions are necessary to represent the data with a minimum of misrepresentation. We can see visualization of this in the figure next to this. It shows that a single dimensional model would represent 33% of the data correctly. Adding a second dimension improves the correctness of the fit by 30% to 63%. Adding a third dimension improves the correctness with another 20% to 83%. The fourth dimension allows one to correctly represent all the data.

The high level of variance (i.e. the fact that the first dimension only represents 33% of the variation) implies that rather than thinking of the color wheel as a low dimensional structure with considerable overlap between the colors we should think of the color wheel as four-dimensional space with each color having its own distinct identity. Therefore we are going to look at all four dimensions.

Rplot2 copy 2Here we can see the first two dimensions. These explain more then 60% of the differences between the colors. The horizontal dimension represents the difference between white and green. There is a set of keywords that occur more often on white cards, such as vigilance, exalted, banding. Green also has its niche with reach, fight and trample. In this structure, blue is quite similar to white (because they both tend to use flying for instance) and red is more similar to green (sharing bloodthirst and trample for instance). Black takes quite a centrist position. The first dimension clearly upsets the color wheel. In terms of the differences in keywords the largest difference is between green and white. There are very little white cards with ‘green’ keywords such as fight (0) and very little green cards with ‘white’ keywords such as lifelink (4). Still there are some signs of color wheel structure with blue being similar to white and green being similar to red.

On the vertical dimension we can see a clear difference between all the other colors and red. Red has its own identity with abilities such as haste and menace. It has some abilities that overlap with white (such as first strike) and some that overlap with green such as bloodthirst.

Rplot3 copy 2 When we add third dimension we can see a similar thing as in the previous figure. There is a strong difference between all the colors and black. On the vertical dimension we can see keywords like fear, intimidate and deathtouch that are characteristically black. Vigilance and fight are characteristically non-black. Of the four remaining colors, blue is most similar to black in terms of this dimension. Blue is an allied colour of black.

Rplot4 copy The fourth dimension separates blue from the other colors. Here typically blue keywords such as scry, shroud, hex proof and prowess are concentrated on the top of the figure. Below one can see typically non-blue key words, such as lifelink and fear. What is interesting about this dimension is that red and green are more similar to blue than white and black, despite the fact that white and black are ‘allies’ of blue and red and green are ‘enemies’

What conclusion can we draw from this?

First, if the color wheel would have relied on strict similarities between allied colors and strict differences between enemy colors, one may have expected a two-dimensional structure where the different colors are ordered in a nice pentagonal structure. This is not the case. Rather than having overlapping keywords with allies, we have high-dimensional space, this implies that every color has its own niche. The colors are about equally distant from each other.

This might simply imply that every color has its own distinct identity and that there is no wheel behind them. But this would be the wrong conclusion: rather than sharing the same keywords (and therefore the same strengths) as their allied colors, the allied colors are likely to compensate each other: for instance, adding white to a green deck might give it the evasion green lacks.

Moreover, this shows that rather than being limited to having an overlap with their allied color, the opposite colors actually overlap as much. Black and green for instance share regenerate and deaththouch. This represents their duality as colors of life and death.

Below one can find the three remaining figures that represent the other perspectives on the four dimensional space.

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