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Mathmatical/geological/astronomical Alignments


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Does anyone know if there's any relevance to a triangle with 90 degrees, 60 degrees and 30 degrees?
It is exactly half of an equilateral triangle. This explains the 2 to 1 ratio of the longest and the shortest side. An equilateral triangle also comes into the vesica piscis, so you could maybe work it somehow into your plot. How precisely did you measure those triangles?

 

A rule and a compass are the best tools for experimenting with these things on paper. For practice and to get the idea, draw a straight line and prick it around middle with the needle of the compass, then find an aperture that doesn't go past either end (but uses most of the length). Trace the semicircle with just a slight excess, so as to clearly mark the pair of points (where it crosses the straight line) and call them A and B. Without changing th e compass aperture, put its needle in one of those points and mark the point C where the new arc crosses the semicircle. Clearly, the triangle ABC is inscribed in the semicircle and has the 2 to 1 ratio, so it is 30° 60° 90° by construction, with the right angle being the one at C. Note: if you trace the complete circle for both needle positions, you get the intersecting circles of an exact vesica piscis!

 

Now, if you have three points and they seem to form that kind of triangle, you can invert the above process more or less. Draw the longest side with the rule, prick the end nearest the third point and adjust the compass aperture to meet it exactly. Then mark where this crosses the line and prick that point to check how exactly the same aperture meets the other end of the long side (which tells you if it is exactly twice) and also how exactly it meets the third point (which tells you how exactly the triangle is inscribed in a semicircle).

 

Is there any relevance to the numbers sequence 13,36,66 ?

 

That's the total length of each triangle

That depends on the scale of the map, so it isn't as interesting as angles or ratios of distances.
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