Fibonacci Sequence & Circles Explained

[brunardot]

Fibonacci sequence and circles

The Fibonacci sequence is a portion of the "revised" Fibonacci sequence.

Five terms of the revised Fibonacci sequence can be found within five different structural parts of a circle.

What are the structural parts of said circle?

 

[Jimmy Snyder]

The Fibonacci sequence is a portion of the "revised" Fibonacci sequence.

What is the revised Fibonacci sequence?

 

[brunardot]

Revised Fibonacci sequence

What is the revised Fibonacci sequence?

Thanks for asking a direct question. I'm used to questions regarding my intelligence rather than questions regarding my statements.

The revised Fibonacci sequence is the first unending Natural integer sequence of the unending, simple additive, sequences of the Brunardot Series, which series is ubiquitous throughout all phenomena:

x, x^2 - 1, x^2, 2x^2 - 1, 3x^2 - 1...

With the above, Brunardot Series, the answer to the initial question should be evident.

 

[xJuggleboy]

I still don't understand this "revised" sequence... Could you please explane it a bit?

 

[brunardot]

1, 0, 1, 1, 2...

I still don't understand this "revised" sequence... Could you please explane it a bit?

Thanks for asking a direct question. I'm used to questions regarding my intelligence rather than questions regarding my statements.

The revised Fibonacci sequence is the first unending Natural integer sequence of the unending, simple additive, sequences of the Brunardot Series, which series is ubiquitous throughout all phenomena:

x, x^2 - 1, x^2, 2x^2 - 1, 3x^2 - 1...

With the above, Brunardot Series, the answer to the initial question should be evident.

The revised Fibonacci sequence (one or two additional terms, depending upon your definition of FS, at the beginning) is the first unending Natural integer (x equals One, "1") sequence of the unending, simple additive, sequences of the Brunardot Series, which series is ubiquitous throughout all phenomena:

x, x^2 - 1, x^2, 2x^2 - 1, 3x^2 - 1...

(Thus, 1, 0, 1, 1, 2... The first term is the revision; so as to, correlate with Nature. This would also pin the unrevised FS as beginning with 0, 1 and not 1, 1)

With the above, Brunardot Series, the answer to the initial question should be evident. (Hint: Consider the circle as a special ellipse.)

 

[Jimmy Snyder]

1, 0, 1, 1, 2

1 - the radius
0 - the focal length *
1 - the major axis *
1 - the minor axis *
2 - the diameter

where * means "Treating the circle as a special case of an ellipse"

 

[brunardot]

Not quite correct

1 - the radius
0 - the focal length *
1 - the major axis *
1 - the minor axis *
2 - the diameter

where * means "Treating the circle as a special case of an ellipse"

No; though you've got the idea. Your answer is contrived, in mixed terms, to fit the sequence.

I don't see how the major axis and the diameter can have different values?

The answer is best in elliptical terms, such that each term represents a different elliptical, structural part, that applies to all ellipses (the parts, not the values) as well as the special ellipse.

Not sure if I've helped or confused?

 

[Jimmy Snyder]

The answer is best in elliptical terms, such that each term represents a different elliptical, structural part, that applies to all ellipses (the parts, not the values) as well as the special ellipse.

Oops, my education did not include definitions of the structural parts of an ellipse. Here is what I should have written:

1 - the radius
0 - the distance between the two foci *
1 - the distance from the center to the ellipse along the minor axis *
1 - the distance from the center to the ellipse along the major axis *
2 - the diameter

where * means "Treating the circle as a special case of an ellipse"

Note that I switched 'minor' and 'major' from what I had originally posted.

You can translate this into the proper structural parts.

 

[brunardot]

Solution accepted

Oops, my education did not include definitions of the structural parts of an ellipse. Here is what I should have written:

1 - the radius
0 - the distance between the two foci *
1 - the distance from the center to the ellipse along the minor axis *
1 - the distance from the center to the ellipse along the major axis *
2 - the diameter

where * means "Treating the circle as a special case of an ellipse"

Note that I switched 'minor' and 'major' from what I had originally posted.

You can translate this into the proper structural parts.

I'll accept your solution, as my given statements are difficult to construe.

I would prefer:

1 = perigee
0 = soliton (half a wave or half the focal length)
1 = vector (line from a focus to the end of the minor diameter)
1 = apogee
2 = major diameter.

You succeeded because you first asked a pertinent question that most assume as irrelevant.

Few persons are educated in the structural parts of an ellipse. Do you realize that every elliptical shape has all its structural parts algebraicly related in the same manner as every other elliptical shape. That is: there is a constant for ellipses as their is "pi" for circles.

Now, find the exact Golden Ratio within said circle.

 

[Jimmy Snyder]

I'll accept your solution, as my given statements are difficult to construe.

I would prefer:

1 = perigee
0 = soliton (half a wave or half the focal length)
1 = vector (line from a focus to the end of the minor diameter)
1 = apogee
2 = major diameter.

I prefer my own answer because it is difficult to misconstrue. What do you mean by soliton? What do you mean by half a wave?

 

[brunardot]

Solution now rejected.

I prefer my own answer because it is difficult to misconstrue. What do you mean by soliton? What do you mean by half a wave?

I have reposted your answer below for ease of reference.

1 - the radius
0 - the distance between the two foci *
1 - the distance from the center to the ellipse along the minor axis *
1 - the distance from the center to the ellipse along the major axis *
2 - the diameter

where * means "Treating the circle as a special case of an ellipse"

Note that I switched 'minor' and 'major' from what I had originally posted.

Now that you have clarified your position, I admit to being overly generous. And, I have, thus, decided to reject your solution for the following three reasons:

1.) The radius of a circle is ambiguous when referring to a special ellipse.

2.) ”The distance between the two foci” is wrong. To be consistent with the analogy, it should be: half the distance between the foci.

3.) Your subsequent switching of the minor and major axes was a good move. However, neither is correct. To be consistent with the analogy, the first should be: the distance from a focus and an end of the minor diameter. The second should be: the apogee (the distance from a focus to the furthest end of the major diameter)

I prefer to think of the focal length as a wave length; therefore the term “wave”; thus, half a wave (the distance from the center to a focus) is referred to as a: soliton. What was not clear in my prior post other than your confusion?

For your convenience:

Brunardot@Brunardot.com
”Click” to directly E-mail Me

 

[Jimmy Snyder]

I prefer to think of the focal length as a wave length

Can you define the focal length of an ellipse for me please?

 

[brunardot]

Sorry

Can you define the focal length of an ellipse for me please?

Sorry, I forgot that there were other interpretations.

I was referring to the distance between the foci.

Sorry, if I confused.

 

[Jimmy Snyder]

1 = perigee
0 = soliton (half a wave or half the focal length)
1 = vector (line from a focus to the end of the minor diameter)
1 = apogee
2 = major diameter.

Are you saying that for the general ellipse:

vector = perigee + soliton
apogee = soliton + vector
major diameter = vector + apogee

 

[brunardot]

?

Are you saying that for the general ellipse:

vector = perigee + soliton
apogee = soliton + vector
major diameter = vector + apogee

You're very good at cutting to the crux.

Actually, the major diameter equals 2v or p + a.

 

[Jimmy Snyder]

You're very good at cutting to the crux.

Flattery will get you nowhere. Please answer the question.

 

[brunardot]

??

Flattery will get you nowhere. Please answer the question.

What part of the below answer do you not understand?

"Actually, the major diameter equals 2v or p + a."

It was a statement concerning the situation; not intended as flattery.

 

[Jimmy Snyder]

major diameter equals 2v or p + a

Yes, I am aware of this equation. But repeating this equation does not answer my question. I was asking if the following is true:

major diameter = vector + apogee

I would expect an answer along the lines of either yes, or no.

I am also asking if the following are true:

vector = perigee + soliton
apogee = soliton + vector

Yes or no would do nicely here as well. My questions are not arbitrary, I want to know because you seem to be calling this sequence a revised Fibonacci sequence (post #9 in this thread):

perigee, soliton, vector, apogee, major diameter.

Did I miss your meaning about this sequence?

 

[brunardot]

I believe I understand the confusion

I believe that now I understand the confusion.

Yes, I am aware of this equation. But repeating this equation does not answer my question. I was asking if the following is true:

major diameter = vector + apogee

I would expect an answer along the lines of either yes, or no.

No.

I stated: "the major diameter equals 2v (2 times vector) or p + a (I usually refer to the apogee as "o" because "a" is the amplitude; thought this would confuse, so used "a"; however, with that: p + o, the perigee plus apogee equals the major diameter."

I am also asking if the following are true:

vector = perigee + soliton
apogee = soliton + vector

Yes or no would do nicely here as well.

Yes.

When I made no mention, I had intended to indicate these were correct.

My questions are not arbitrary, I want to know because you seem to be calling this sequence a revised Fibonacci sequence (post #9 in this thread):

perigee, soliton, vector, apogee, major diameter.

I appreciate your questions; did not intend to imply otherwise. You have ventured where no other would.

I was not clear; the above is only true for the special ellipse, which is the circle. Only the first four terms of RFS apply to regular ellipses.

The confusion was entirely my fault. To mitigate, I have little time and monitors from other threads on my butt all day.

Did I miss your meaning about this sequence?

See my last sentence above.

 

[Jimmy Snyder]

the above is only true for the special ellipse

The same can be said for my original answer (post #8 in this thread), the one you tentatively accepted and subsequently rejected:

1 - the radius
0 - the distance between the two foci *
1 - the distance from the center to the ellipse along the minor axis *
1 - the distance from the center to the ellipse along the major axis *
2 - the diameter

where * means "Treating the circle as a special case of an ellipse"

 

[brunardot]

Yes. but . . .

The same can be said for my original answer (post #8 in this thread), the one you tentatively accepted and subsequently rejected:

1 - the radius
0 - the distance between the two foci *
1 - the distance from the center to the ellipse along the minor axis *
1 - the distance from the center to the ellipse along the major axis *
2 - the diameter

where * means "Treating the circle as a special case of an ellipse"

When I first commented "No," I did not like your order of the major and minor diameters. I felt they were contrived to match the integers rather than their ascending order. (You subsequently, reversed them.)

I, also, did not like your mixing two circular terms (radius and diameter) with two elliptical terms (major and minor diameters.)

I felt there was some confusion, on your part, as to what were the corresponding circular and elliptical parts. This correspondence is more salent to general applications of the Brunardot series to ellipses; than, the RFS to circles.

The RFS is also a special case of the more general BS.

Regardless, I now feel you have a grasp of the underlying principles of the problem.

Are you ignoring; or, have you missed my earlier relation to the Golden Ratio?

 

[Jimmy Snyder]

When I first commented "No," I did not like your order of the major and minor diameters. I felt they were contrived to match the integers rather than their ascending order. (You subsequently, reversed them.)

I, also, did not like your mixing two circular terms (radius and diameter) with two elliptical terms (major and minor diameters.)

Did not like? That is irrelevant. My post was a correct answer to the puzzle.

 

[brunardot]

I disagree

Did not like? That is irrelevant. My post was a correct answer to the puzzle.

I disagee; as you also have the right to.

I suspect we will need Korzybski (now dead) to referee.