TO DIVIDE AN ANGLE IN ANY NUMBER OF EQUAL PARTS
shyamal kumar das
majo...@mathforum.org
subscribe geometry-research
Respected Sirs,
I would like to inform you that most probably I have invented a geometrical method / construction, by which any angle can be divided into any number of equal parts.
So far I know, a few people had tried to trisect an angle, but failed to do so. My method can not only trisect but divide any angle into any no. of equal parts.
Please be kind enough to look into my method. If you find it all right and worth publishing in your journal /magazine, kindly bring out the same into the limelight.
I am an ordinary man of low profile and little knowledge. I do not know what to do and how to complete the formalities before submitting the write-up.
Now , for the sake and benefit of all students who know how to divide a straight line but do not know as yet how to divide an angle into any no. of equal parts , kindly publish my method worldwide without further delay .
I solicit an acknowledgement from your end.
Thanks and Regards,
Very Truly Yours,
SHYAMAL KUMAR DAS
Address: Flat No: 1 C/ 15
Uttarpara Housing Estate
88B, G. T. Road
P. O.: Bhadrakali, Uttarpara
Hooghly, West Bengal
Pin code no: 712232. INDIA.
Res: (033)26645991
Mob: 9432861354
EMail: das.sh...@gmail.com
Attachment available from http://mathforum.org/kb/servlet/JiveServlet/download/130-2096846-7132870-607827/file3.pdf
Peter Scales
> Respected Sirs,
>
> I would like to inform you that most probably I have
> invented a geometrical method / construction, by
> which any angle can be divided into any number of
> equal parts.
> So far I know, a few people had tried to trisect an
> angle, but failed to do so. My method can not only
> trisect but divide any angle into any no. of equal
> parts.
>
> Please be kind enough to look into my method.
Sir,
The method you propose does give a reasonable approximation to the 1/n th value of an acute angle for reasonably large n, but unfortunately it is not a valid method for exact angle division.
The flaw is that it equates the base (chord) of an isosceles triangle with the corresponding arc of a sector, which is simply not true.
This leads to an error in your construction. When you use a compass to lay off a length, it is a chord you lay off, not an arc. There is no unique arc associated with a chord, and in your example two different arcs occur: arc DM with r=1 and arc HI (=IJ etc) with r=5. The chords are identical, but the arcs are not.
Your comment on page 3 regarding 60degrees being an approximation for 1 radian arises from the same error. In an isosceles triangle, a side of unity subtends 60 degrees at the apex whereas an arc of unity subtends 1 radian = 57.296 degrees. This is not an approximation, but a different value from a different figure.
As an example, use your method to bisect a right angle:
BD=1, chordBE=sqrt2, BA=2, CF=FG=sqrt2
: your approx to the bisected angle, theta, has
sin(theta/2)=1/4*sqrt2 .: theta/2=20.7deg theta=41.4deg
and the discrepancy is 90-2*41.4=7.2deg
It is even more obvious if you use your method to bisect a straight angle=180 BD=1 DE=2 and the resultant approx bisected angle is 60deg, not 90deg.
I hope this has been of some help to you.
Since you have access to the Internet you could Google angle division and explore the extensive existing literature.
Regards, Peter Scales.
shyamal kumar das
>From :Shyamal Kumar Das
Sir,
Hope, you have recd my two letters dated Aug,11 and Aug 13 on the subject. But, I have not recd any reply from you.Pl. answer me and oblige. Tkx & Rgds
Peter Scales
Sir,
To my way of thinking the academic or intellectual merit of any procedure derives from its correctness, its inherent accuracy or its ingenuity. If the result is exact, simplicity, ingenuity or minimum and least complicated operations are factors in establishing relative merit. If the result is an approximation, then similar factors determine its usefulness, generally measured as the accuracy achieved divided by the number of operations required to achieve it.
Depending on the use envisaged for the method, one might add ease of understanding and remembering it and relative simplicity of operations if the method yields acceptable accuracy for student or artisan use.
You say: "However, my intention is to develop an easy and simple method for the students to divide an angle
along with the method to divide a straight line ,the later is known to them."
When I first read your post I was unsure of the claims being made and of the target audience. In one sense I thought it was being proposed as an exact method, even though the write-up included hints on how to improve the accuracy. To my initial understanding these hints semed more directed at overcoming practical inaccuracies, rather than being an inherent part of the underlying theory. This was reinforced by the comments on 60deg approximating 1 radian.
However the above comments are now history and relate only to my initial response, which I hoped would be helpful to you.
Your more recent post clarifying that the method is proposed as a practical method for students, easy to understand and easy to implement, which gives reasonable accuracy for acute angles.
Provided the inherent flaw in the method is understood, it should be useful for student use. I did indicate this in the opening sentence of my first post.
I hope this clears up any misunderstanding,
Sincerely, Peter Scales.
shyamal kumar das
FSir,
While I thank you for your letter posted on sept 1, 2010 on the subject, I fail to understand whether the same is your concluding letter, saying: so,I reject your stupid method and, don’t disturb me further with your idiotic ideas.
I am extremely sorry for the mistake I made regarding 60 degree and 1 radian. My apology to you for that mistake.
In my previous letter posted on august,11 on the subject, I cited another example i.e, “pai”. You have not enlightened me on that. Please do,if your time permits, to clear the misunderstanding regarding “pai”= 22/7 or, 3.14 which is acceptably accurate, while, my method of dividing an angle after bisections for perfection which is not accepted.
If my method is 100% perfect, the question of improvement for accuracy would not arise, I would not have written and elaborated under a separate column : Certain constraints and its probable solutions. Not only that ,on page 2 of my New Theorem (proposed), in last paragraph under Remarks column , how to better the process to get acceptable accuracy. There, I wrote : “ Please note that more bisection, more accuracy , if required “.
Surprisingly, you have made no remarks, neither on bisections prior to division, written in my original write-up, nor, on dividing 180 degree into five equal parts, mentioned in my letter posted on 11,Aug. I would like to know how much it is far from the acceptable accuracy. If required, the angle 22.5degree can further be bisected to get 11.25degree, before actual division into five equal parts.
I mentioned about practical inaccuracies as first constraint, because construction comes first where students make mistakes in drawing. However, my method is not only for acute angle it is also suitable for obtuse angle and reflex angle. To justify that , I took the case of 180 degree to be divided into five equal parts.
To conclude, I would humbly request you to go through and not to glance through all my letters and write-up once again with a positive frame of mind. I don’t want to be Saurav Ganguly andyou,my coach Mr. Chapel. Thanks and Regards. Shyamal Kumar Das. Sept. 2nd,2010
ROM: Shyamal Kumar Das
Peter Scales
> While I thank you for your letter posted on sept 1,
> 2010 on the subject, I fail to understand whether the
> same is your concluding letter, saying: so,I reject
> your stupid method and, don’t disturb me further with
> your idiotic ideas.
I hope I did not really give that impression!
> I am extremely sorry for the mistake I made regarding
> 60 degree and 1 radian. My apology to you for that
> mistake.
My comment was to show the difference between the angle subtended by a chord, and that subtended by an arc.
> In my previous letter posted on august,11 on the
> subject, I cited another example i.e, “pai”. You have
> not enlightened me on that. Please do,if your time
> permits, to clear the misunderstanding regarding
> “pai”= 22/7 or, 3.14 which is acceptably accurate,
Pi is the ratio of Circumference/Diameter. It is neither 22/7 nor 3.14 for theoretical considerations or proofs, but these values may be useful in practice for approximating calculated values.
> If my method is 100% perfect ... I would
> not have written ... : Certain constraints and its
> probable solutions ...(or) ...how to better the
> process to get acceptable accuracy. ... more
> bisection, more accuracy , if required.
I did acknowledge your method for acute angles, and I understand your approach of subdividing larger angles into smaller appropriately acute angles, but each subdivision, and later expansion, involves more operations, thus reducing the accuracy per operation achievable.
> To conclude, I would humbly request you to go through
> and not to glance through all my letters and write-up
> once again with a positive frame of mind.
It was not my intention to be critical of your method. It seems easy to understand and remember for students and artisans.
When I was at school we would construct an n-section of an angle by trial and error. Not very sophisticated, but quite practical, and not involving many operations to achieve drawing accuracy. You have given the process more structure, and that should be a help to your students provided they understand the underlying theory, incuding why it is necessary to subdivide an obtuse angle, and the difference between the accuracy theoretically achievable and that achievable in practice with a compass on paper.
I hope my comments have been of some help, and that I have now cleared up any miunderstandings.
Sincerely, Peter Scales.
shyamal kumar das
From: Shyamal Kumar Das
Sir,
I thank you once again for allowing me and encouraging me to think over further regarding Arc length vs Chord length to minimize the difference and to get acceptable accuracy.
Here are some outcomes:
1 ) when the arc length is equal to the radius of the arc ,the angle subtended at the center = 1 Radian = 57.3 degree again, when the chord length is equal to the radius of the arc, the angle subtended at the center= 60 degree Therefore, when angle subtended at the center = 57.3 degree, arc length minus chord length corresponds to the difference of ( 60—57.3)= 2.7 degree So, for 22.5 degree subtended angle, the difference = (2.7/57.3) *22.5 =1.06degre = 4.7% Hence, error accrued = 1.06 degree = 4.7%
2 )for 60 degree subtended angle and R being the bigger radius = nr, where n = no. of parts and r = smaller arc radius, Arc length minus Chord length = R*pi/3 -R =R (pi/3 - 1) = R (22/21 - 1)= R/21 so for22.5degree subtended angle, the difference =R/21 * 22.5/60 =1/56 of R = =1.8% Hence, error accrued = 1.8% of R. If we take n=no of division =5and r =small arc =0.5 cm, then, R =n.r = 5*0.5 =2.5 cm, error accrued = 1.8% of 2.5 cm = .045cm
Regarding the no of operations involved, let us take the example of 180 degree to be divided into 5 no of divisions .
a) 3 bisections involve 3operations
b) One sector of 22.5degree to be divided into 5 parts to get 5 divisions – requires 1 operation
c) Taking one small div., or one graduation i,e, 1/5 of 22.5 degree , another 3 graduations to be added to get 8 graduations, means 1 operation
d) Taking 8 graduations at a time, whole arc of 180 degree is to be divided into 5 equal parts is the sixth operation
Hence, dividing into 5 equal parts for an obtuse angle involves 6 operations, while dividing any straight line into 5 equal parts requires at least 5 operations.
Please correct me if I am wrong in my above observations. Otherwise, please let me know if the errors accrued and no of operations involved are within the acceptable accuracies.
I am now 65 years old . I don’t know how do you like to teach an old man like me. But, believe you me ,pleasure is mine. Regards .Shyamal Kumar Das.
shyamal kumar das
From: Shyamal Kumar Das
Sir,
Now, I am writing you this letter because of the following reasons:
1) I ‘ve realized that again I ‘ve made mistake applying unitary method which is not correct . Again sorry sir.
2) Any way, what is very much annoying me is the drawback of my method. The difference between arc length and chord length and how to overcome that have spoilt my sleep at night.
3) I ‘ve calculated out something which ,according to me, is very much interesting and
relevant.
CALCULATION :
A) When , the angle subtended at the center= 180 deg.,
The difference between arc length and chord length= pi*R-2R=R( pi-2 ) =R ( 3.14-2) = 1.14R
B) When , the angle subtended at the center= 90 deg.,
The difference between arc length and chord length= pi/2*R- v2R= R( 1.571-1.414)= 0.16R
C) When , the angle subtended at the center= 60 deg.,
The difference between arc length and chord length= pi/3*R- R = R( 1.047-1 ) =0.047R
D) When , the angle subtended at the center= 30 deg.,
The difference between arc length and chord length= pi/6*R-2Rsin15 deg
= R(0.5233- 2*0.2588) = =0.0057R
E) When , the angle subtended at the center= 22.5 deg.,
The difference between arc length and chord length= pi/8*R- 2Rsin11.25deg =R(0.3925- 2*0.1951)= =0.0023R
F) When , the angle subtended at the center= 1/5 of 22.5deg = 4.5deg
The difference between arc length and chord length= pi/40- 2Rstn2.25deg
=R( 0.0785-2*0.0393 ) ???
OBSERVATION & INFERENCE :
Hence, it is observed that, when angle subtended at the center tends to zero, the difference between arc length and chord length also tends to zero. Thus, we can infer that when angle subtended at the center is 4.5deg, the arc and chord lengths are almost same. In my previous letter where I took the example of 180deg., to be divided into 5 equal parts, the cut-off arc / chord length has been taken for graduation, against subtended angle = 4.5deg. at the center.
Please correct me ,if I am wrong. I am eagerly waiting for your reply on this. If you are O.K with my calculation, observation and inference, then I shall only hope that you will approve my method which is almost correct. Thanks and regards. Shyamal Kumar Das 09.10.2010
Peter Scales
> =R(0.3925- 2*0.1951)=
> =R(0.3925- 2*0.1951)= =0.0023R
> F) When , the angle subtended at the
> center= 1/5 of 22.5deg = 4.5deg
> The difference between arc length and chord length=
> pi/40- 2Rstn2.25deg
> =R( 0.0785-2*0.0393 ) ???
> OBSERVATION & INFERENCE :
> Hence, it is observed that, when angle subtended at
> the center tends to zero, the difference between arc
> length and chord length also tends to zero. Thus, we
> can infer that when angle subtended at the center is
> 4.5deg, the arc and chord lengths are almost same.
> In my previous letter where I took the example of
> 180deg., to be divided into 5 equal parts, the
> cut-off arc / chord length has been taken for
> graduation, against subtended angle = 4.5deg. at
> the center.
> Please correct me ,if I am wrong. I am eagerly
> waiting for your reply on this. If you are O.K with
> my calculation, observation and inference, then I
> shall only hope that you will approve my method which
> is almost correct. Thanks and regards. Shyamal Kumar
> Das 09.10.2010
Sir,
Sorry to be tardy in replying. It is not due to lack of interest, but pressure of other duties!
The accuracy you need to achieve depends on the use to which you intend the method to be applied.
I had thought that the following two benchmarks may be useful in your assessment:
1. For student or artisan use, with a school-type compass and reasonably sharp pencil, to obtain a result within 1/4 of a degree, and
2. For precise engineering drawing on good quality paper with a professional compass and a hard, very sharp pencil to obtain a result within 1/10 of a degree.
Give these numbers some thought. You may think they are a bit too precise. The matter is open to debate.
You might like to experiment and see what accuracy you can achieve. Each time you place the compass point, or scribe an arc, you introduce some small error. Random errors tend to cancel out. Systematic errors tend to accumulate.
Eventually you should be able to refine your rules of application to achieve some stated accuracy.
Then it will be up to users to decide whether your method is more or less useful in practice compared to other methods that are available.
Regards, Peter Scales.
shyamal kumar das
From:Shyamal Kumar Das
Sir,
I consider myself hopeless and helpless. This is unfortunate.=20
Your letter seems to me ,very irrelevant to my last letter on the subject. =
Our two letters had two different statements altogether. While, I am trying=
to do something to minimize the difference between arc length and chord =
length which is the main criteria to be considered, you are pointing out th=
e different constraints of the students, artisans ,engineers with their t=
ools to draw the geometrical constructions. For their personal error ,whet=
her big or small, whether random or systematic with their ordinary or soph=
isticated tools, it is not the inventor to be blamed. If a student fails to=
achieve 100% marks no teacher or mentor is punished for that.=20
I am here to concern with the drawback of my method . My objective is to d=
erive some solution. In my last letter post on oct 8,2010, I =E2=80=98ve gi=
ven my findings, detailed calculation, then my observation and inference th=
ere for your kind perusal , which most probably escape your attention for =
your pre-occupation and tight schedule.=20
Now, I am happy to state that I =E2=80=99ve found out the solution and my =
objective is fulfilled. My calculation reveals that when angle subtended at=
the center is 4.5 deg, the result is not even within 1/10 deg,it gives ze=
ro error. In other words, 4.5 deg, subtended angle gives not only appreci=
able accuracy but accurate result. And, please note, I repeat, please note =
that the graduation had been made when the angle subtended at the center i=
s 4.5 deg. and given angle is 180 deg, to be divided into 5 equal parts. Al=
ong with this letter I am sending the revised version of my THEOREM for yo=
ur kind perusal and approval, if you please.=20
I understand from the last paragraph of your letter that other methods are =
available. Kindly let me know those along with their merits and demerits. I=
f you think anyone is better than that o f mine, pl., confirm. I =E2=80=98=
ll not disturb you anymore.Sincerely, ShyamaKumar Das 12-10-2010
Attachment available from http://mathforum.org/kb/servlet/JiveServlet/download/130-2096846-7238725-634697/NEW%20THEOREM%20(Revised%20).docx
shyamal kumar das
FROM :Shyamal Kumar Das
Sir,
Please replace the third page of my proposed theorem ( revised )with the page attached now. sorry for the trouble. thanks and regards.
Attachment available from http://mathforum.org/kb/servlet/JiveServlet/download/130-2096846-7244089-635800/CERTAIN%20CONSTAINTS%20AND%20ITS%20POSSIBLE%20SOLUTIONS_2.doc
shyamal kumar das
>From :Shyamal Kumar Das
Please try yours or anybody's method to trisect or divide 180 degree into five equal parts to justify your statement that my method is not new or unique . I am 65 years old but still ready to learn from others and appreciate other's work. best wistes. Shyamal kumar Das. 04.11.2010