Construction of a pentagon using only the grid system

Join the discussion
Ask a follow-up here, or get your own question answered by working scientists, mathematicians and engineers — people, not an autocomplete.
Real named experts · corrections over time · the nuance an AI answer skips
Sign up Log in
9 replies · 3K views
Scotti G
Messages
3
Reaction score
0
I have found a way to construct a pentagon using only the grid system. The internal angles are all within 99% accuracy and all the angles to the 3rd decimal add up exactly to 540 degrees. This is without the use of a compass and bearing in mind that the internal lines intersect at the ration of phi, it is theoretically impossible to construct a perfect pentagon. Is this "discovery" of any value or consequence?
 
Mathematics news on Phys.org
If we are talking about the same thing, I think this is something I tried to do as well. I started with the rules for a compass and straight-edge construction of a pentagon. Then I put the apex on the y-axis and the two adjacent vertices on the x. Then I 'translated' the construction rules into triangles, and then I just went to town pythagorean style. My goal was to do it without using trig. Is this something like what you did?
 
Scotti G said:
Is this "discovery" of any value or consequence?

Not really. It's a neat curiosity though. Maybe you'll even get it published in a journal on Euclidean geometry. But these kinds of constructions (especially approximate ones) haven't been important to mainstream mathematics for decades.
 
  • Like
Likes   Reactions: mfb
Scotti G said:
The internal angles are all within 99% accuracy and all the angles to the 3rd decimal add up exactly to 540 degrees.
Internal angles of any 5 edged polygons (pentagons as well as any iregular 5 edged shape) will always add up to precisely 540 degrees.

http://www.mathsisfun.com/geometry/interior-angles-polygons.html
 
  • Like
Likes   Reactions: Drakkith
Terribly sorry to mention it, but this thread is not about constructing a pentagon using a straight edge and compass. It is (I think) about identifying the Cartesian coordinates which would lie on a circle, and would correspond to the vertices of a regular pentagon without the use of trig.

Of course, for all I know, it could be ridiculously simply to do what I described and Scotti G might be talking about something completely different. But the OP has not checked in since posting, so no clarification on that score yet.
 
Yes my goal was to construct a pentagon without trig or the use of a compass. What was the accuracy of you angles with your construction? If you say you
started with the rules for a compass and straight-edge construction of a pentagon then you "tried" it without trig but not without a compass.
On a different subject I think my "discovery" could very well be used in nature to construct pentagonal structures such as starfish or flowers as the construction using only a grid system is far more plausible than the use of trig or a compass at atomic level.
 
I didn't really get that far. And now I'm confused by what you mean by 'without a compass.' Any chance you could share what you did? At least describe it in more detail?
 
I simply used only the grid system. No trig. no compass. no protractor. " I started with the rules for a compass and straight-edge construction of a pentagon" - you. I started only with the grid and my pip. It took 14 years of persistence but the construction only has 7 lines and you have a pentagon with all the angles within 99% accuracy, to be precise the lowest accurate angle shared by 2 angles is 99.39%. I don't believe a more accurate pentagon could be constructed using only the x and y axis.