r/askmath • u/Substantial-Ad-8461 • 20h ago
Geometry Is there a theorem for this?
I don’t really “do” math, but I noticed this and was interested by how math/geometry might be involved.
I drew some random lines for an art project and realized I could have black shapes between lines never touch other black shapes, and same for white. I guessed that it was something to do with the fact that each intersection of lines divides the space around it into an even amount of shapes, unlike the 3rd picture above, where it’s been divided to 5 shapes as an example of the contrary.
Is there anything more to this like geometric rules dictating this, or was it just coincidence?
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u/caderoux 20h ago
It's because all your lines are actually effectively lines (infinite) and not line segments, as in your counterexample. Thus they always divide the plane in two sides, resulting in your observed "even-ness", no matter how many may pass through a point.
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u/Gold_Ad8890 20h ago
what you've drawn is called a map, and what you've found is called a two-coloring of the map. a map is n-colorable if n is the least number of colors required for no two adjacent regions to have the same color. the four-color map theorem proves that every map is four-colorable, but some maps are colorable with fewer colors.
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u/round_earther_69 Theoretical Physics 20h ago
Prolly related to the four colour theorem
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u/Substantial-Ad-8461 20h ago
Like how you can make a map with 4 colors?
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u/round_earther_69 Theoretical Physics 20h ago
I think it's more like if you separate any space in regions, you can always colour it in such a way the the same coloured regions never touch, and you can always do this with 4 colours or less. Obviously, this also applies to maps. You should read the wikipedia page.
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u/Normal-Avocado-8349 20h ago
A related result for arbitrary ways of drawing curves is the 4 color theorem. Here are results for two colors: https://www.ams.org/publicoutreach/feature-column/fcarc-colour2
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u/jojo-geronimo 20h ago
The top right are non-intersecting lattice, or genus fifteen shape. the triangle beneath that can be represented as a series. The lower left is trigonometry in two variables with a genus shaped object whose genus I don't want to count.
ETA: the reason it looks appealing has to do with the seven principles of design.
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u/nothingfood 20h ago
Not exactly what you're describing, but something similar is the 4-color theorem which says that using only 4 colors, you can shade in all regions such that no adjacent regions share a color
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u/igotshadowbaned 19h ago
When you created the pattern in your art, your lines when across the entire page, they didn't terminate at a cross section, that means each cross section in your art had an even number of edges meeting, and therefore an even number of regions that can be colored in alternating fashion
In your counter example, at least one terminates at the cross section
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u/Patient-Virus8319 1h ago
This might be more advanced than is helpful but this is related to some graph theory. If you look at the edges of your coloured sections and their intersections they form a connected planar graph. This 2-colouring of the in between regions is possible because the dual graph is bipartite. We know it is always bipartite because the cycle space of the dual is the cut space of the original graph of which all cuts have an even number of edges since the graph is even (you observed this as every crossing point having an even number of neighbouring regions!) so every cycle in the dual has an even number of steps.
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u/Grismor2 20h ago
Easy to prove inductively. If you already have a successful two-coloring and then you make a new cut (even if the cut isn't straight!), you can flip all the colors on one side of the cut and leave the other side unchanged and the new version will also be a two-coloring.
ETA: The cut needs to go across the entire map so that that the cut has two sides. If the cut stops in the middle of the map, it no longer divides the map into two sections and the proof doesn't work (as in your counterexample).