Announcement

Collapse
No announcement yet.

Circle Division

Collapse
X
 
  • Filter
  • Time
  • Show
Clear All
new posts

  • mklotz
    replied
    I think the attached diagram captures your proposed technique for trisecting an angle. If it doesn't, at least it will suggest a method for proving your technique.

    I've turned things around for the proof. I've started with three equal angles (A in the figure). On the middle angle I've constructed a bisector (PQ length 's') and drawn the line x1-Q-x4 perpendicular to 's' that cuts across all three angles.

    If your technique works, it should be possible to show mathematically that the angles divide the line into three segments...

    a = x1-x2
    b = x2-x3
    c = x3-x4

    that are equal, i.e. a = b = c.

    Aside: To my eye, it already appears that 'a' is longer than 'b', but in math "eyeballing" carries no weight so on with the proof.

    We can solve for 'b' in terms of 's'

    b = 2 * s * tan (A/2)

    We can also solve for:

    z = Px2 = s / cos (A/2)

    w = Px1 = s / cos (3A/2)

    Knowing 'z' and 'w', we can use the law of cosines to find 'a'.

    a = [w^2 + z^2 - 2 * w * z * cos(A)]^0.5

    So, all you need to do now is prove that a = b using the expressions above. If you can then your technique will work. Good luck; be sure to show us your work.

    Click image for larger version

Name:	TRISECT.jpg
Views:	37
Size:	224.1 KB
ID:	1930671

    Leave a comment:


  • mklotz
    replied
    Originally posted by nickel-city-fab View Post

    Notice how I said you draw a chord? making your angle into a triangle. You then divide the chord line into whatever (hell we did this in HS) and draw from those points back into the center of your circle...
    I understand what you said. Now you need to show the math to prove that your method works. A verbal description of what you intend to do isn't enough.

    Leave a comment:


  • nickel-city-fab
    replied
    Originally posted by mklotz View Post
    I'll be looking forward to your demolition of a proof that's survived for 184 years. Be sure to publish it here for all to see.
    Notice how I said you draw a chord? making your angle into a triangle. You then divide the chord line into whatever (hell we did this in HS) and draw from those points back into the center of your circle...

    Leave a comment:


  • mklotz
    replied
    I'll be looking forward to your demolition of a proof that's survived for 184 years. Be sure to publish it here for all to see.

    Leave a comment:


  • nickel-city-fab
    replied
    Originally posted by mklotz View Post
    The statement is "trisection of an angle USING ONLY STRAIGHTEDGE AND COMPASS is not possible", not "trisection of an angle is not possible". AFAIK, the first statement still stands.

    There are probably many ways to trisect an angle.
    -- Indeed -- and the method I mentioned uses only a straightedge and a compass. The worked examples that i have here show "dividing a line into an arbitrary number of equal parts".

    Leave a comment:


  • mklotz
    replied
    The statement is "trisection of an angle USING ONLY STRAIGHTEDGE AND COMPASS is not possible", not "trisection of an angle is not possible". AFAIK, the first statement still stands.

    There are probably many ways to trisect an angle.

    The trisection "tomahawk"...

    https://en.wikipedia.org/wiki/Tomahawk_(geometry)

    is an example of one trisection mechanism, although it cannot be used on certain angles.

    Leave a comment:


  • nickel-city-fab
    replied
    Originally posted by Lew Hartswick View Post
    Was told, as of 1949, by my Trig / Geometry teacher the "angle" couldn't be trisected. Has there been any new construction technique that does it???
    ...lew...
    I've heard the same thing. So I actually went and downoladed Euclid's geometry, the original greek book has been translated. Link: https://archive.org/details/thirteenbookseu03heibgoog

    I've wondered about simply drawing a "chord" across the end of an angle, and dividing the line into whatever number of parts such as 3 parts, like you said, or 7 parts. Instead of being able to do it directly. I think it would work.

    Leave a comment:


  • Lew Hartswick
    replied
    Was told, as of 1949, by my Trig / Geometry teacher the "angle" couldn't be trisected. Has there been any new construction technique that does it???
    ...lew...

    Leave a comment:


  • Paul Alciatore
    replied
    And EXPLAINING it.

    Leave a comment:


  • Paul Alciatore
    replied
    Well, don't let me stop you from showing the proof.



    Originally posted by Stargazer View Post

    Don't mean to be disagreeable but he answer is yes. The Peano axioms and Zermelo-Frankel set theory provide a solid foundation for basic arithmetic. Even if one objects to set theory as a basis for arithmetic foundations, the Peano axioms can also be understood and established by way of category theory.

    Leave a comment:


  • JRouche
    replied
    Originally posted by dian View Post
    mathematics is a theory that cannot be validated, as actually nothing can be validated. as it cannot be excluded that one day you "drop" a stone and it takes off towards the sky (or e.g. disappears).

    right?
    Well, different studies.

    There is Theoretical Math and Real Math. Lol Thank god Einstein cant hear me now.

    Most folks cant delve into the theory part of it. Most can get into the actual numbers of it though. JR

    Leave a comment:


  • JRouche
    replied
    Originally posted by Astronowanabe View Post
    Sure arithmetic as drilled in school helps expressing ideas to others
    and gives you a language/vehicle to learn what others have done.
    But after that math is patterns, and patterns of patterns that just keep going
    till they are completely divorced from our every day real world but
    can still be brought back in a way that matters here.
    Yes Sir, thats what I was trying to say. Math is self solving. If you made a number sequence that wont work you will find it with math. Just go backwards. (oh, then fix it) JR

    Leave a comment:


  • dian
    replied
    thanks, i guess i have to study the quran first.

    Leave a comment:


  • mklotz
    replied
    Originally posted by dian View Post

    sorry, link doesnt work.

    the peano axioms cant prove anything, beause they are, well, axioms. i think paul is right.

    explain to me how it can be excluded, that one day you put an apple into a basket, then put in another one and wind up with three apples (or none)? besides, apples (as stargazer explained) have nothing to do with mathemathics.

    mathematics is a theory that cannot be validated, as actually nothing can be validated. as it cannot be excluded that one day you "drop" a stone and it takes off towards the sky (or e.g. disappears).

    right?
    Here's the link again...

    https://quod.lib.umich.edu/cgi/t/tex...=image&seq=401



    Leave a comment:


  • dian
    replied
    Originally posted by mklotz View Post

    Here you go...a page from Whitehead and Russel's Principia...

    https://quod.lib.umich.edu/cgi/t/tex...aat3201.0001.0 01&frm=frameset&view=image&seq=401

    but be advised, if you're not a professional mathematician and use math more as a tool, your time would be more profitably spent learning some practical math rather than farking about with abstruse, quasi-philosophical stuff like this.
    sorry, link doesnt work.

    the peano axioms cant prove anything, beause they are, well, axioms. i think paul is right.

    explain to me how it can be excluded, that one day you put an apple into a basket, then put in another one and wind up with three apples (or none)? besides, apples (as stargazer explained) have nothing to do with mathemathics.

    mathematics is a theory that cannot be validated, as actually nothing can be validated. as it cannot be excluded that one day you "drop" a stone and it takes off towards the sky (or e.g. disappears).

    right?
    Last edited by dian; 02-21-2021, 05:47 AM.

    Leave a comment:

Working...
X