Discovering the Name and Meaning of a Classic Geometry Diagram [Broken]

[jeffrey c mc.]

I have no doubt that this diagram has a specific name, and that it appears in Geometry texts. I don't now the name, or what text it may be in. It's plane Geometry, and it shows a definite relation of the radius of a Circle to the Line of it's Circumference. I may tie it into the thread on my discussion on calculating the volume of a sphere without using the standard equation or methodology of the classical geometers. Just for something to task on. As the circle [circ.] in the diagram is divided into six equal segments, I believe the illustration is, or may be illustrating, two pi radians; or at least can be described in those terms. I only intuitively understand two pi radians, though, so I may be confused. I have included a link to image, and will also include it as an attachment. I photographed the diagram with a camera phone, with LED flash enabled, and, and fan-dazzle.

Well, the insert image tags seem to not be appearing, hope the attachment appears. I shaded some areas in the inner square for my own purpose.

 

[Simon Bridge]

I only intuitively understand two pi radians

It's not clear what this means - but it is likely that the intuitive understanding could use some concrete reference.

The "radian" is the length of the radius taken as a unit.
There are 2-pi of these in a circumference.

If you make a wheel with a radius of 1 meter, roll it exactly once along the ground (no slipping), then the center travels a distance of (about) 2-pi meters. Pi itself is only found in the limit.

it shows a definite relation of the radius of a Circle to the Line of it's Circumference.

It is not clear how the diagram attached relates the circumference to the radius of a circle.

 

[jeffrey c mc.]

Simon;

Lacking a classical training in higher forms of mathematical education. I have to rely on the instructions of my father and his father. My father was a Structural Engineer, my Grandpa was a Land Surveyor. While the lessons may not have been completely understood while I was being 'schooled'; by the same coin, they may have. Let me describe the graph. Using a fine compass; ie, one with a mechanical adjuster; I inscribed a two inch circle; appx, close enough for my purposes though; Then, where the line of the circumference intersected with the vertical line of the 'graph', using those coordinates as the axis, I used the compass, which remained at the original setting; to inscribe two more circles. Then, starting at the point(s) of intersection, of the original circle, on the vertical; which are two in number; I used the intersection(s);which were formed of the intersections of original circle and of the two separate circles on the vertical axis; these comprised four additional points. Then, when I connected all these points; from point to point; I found that I had drawn a regular hexagon. Which has six equal sides, and six equal angles; and which; the six points fell on the circumference of the circle I used to define the coordinates, thereof. as I used a compass, that was used--unchanged--in the plotting, or graphing; the coordinate points in the two-dimensional--pencil and paper--diagram; I thus perforce proclaim that the correlation of the radius to the line of circumference, is 1 to 6. Okay so from a quick trip to see if this is actually a hexagon, I find this hexagon was described by Euclid and is one of his 'elements' and appears in book four, proposition 15. Doesn't appear to have a name although; as in Euclid's Hexagon. And since, therefore, as Simon did not understand the relation, I do so pronounce that perhaps Euclid did not neither, therefore; I dost pronounce that it shall be known as McMahan's variant to Euclid's 15 proposition, and dost also challenge that personage to defend his proposition; or explain his propositions in a more formidable manner. On guard Euclid, on guard thou noble geomatarian. Heh, spell check wanted me to replace that with vegetarian, can you imagine.

 

[Simon Bridge]

Using a fine compass; ...

I can see the lines for how you constructed the picture. If you'd just filled the page with intersecting circles like the seven you drew, and connected the dots in diamonds, it would be an isometric graph.

...I thus perforce proclaim that the correlation of the radius to the line of circumference, is 1 to 6.

I'm sorry, I don't see why that follows "perforce" from what you drew.
Where did you get that 1:6 ratio from?

If the distance from the center to a corner of the hexagon is 1 unit, then the perimeter is 6 units - this what you mean? But if I set 2 units to be the perpendicular separation of opposite sides I get a different ratio.

It's unclear what you mean by "correlation" too - you really have to stop using technical words in a poetic way when you are trying to make yourself understood. You don't really mean to say that the ratio of the circumference to the radius is 6 do you?

Also - considering it is you who is making the odd claim, it is you who must elucidate and defend it - which is handy since Euclid is unavailable right now ;)

 

[Mark44]

Simon;

Lacking a classical training in higher forms of mathematical education. I have to rely on the instructions of my father and his father. My father was a Structural Engineer, my Grandpa was a Land Surveyor. While the lessons may not have been completely understood while I was being 'schooled'; by the same coin, they may have. Let me describe the graph. Using a fine compass; ie, one with a mechanical adjuster; I inscribed a two inch circle; appx, close enough for my purposes though; Then, where the line of the circumference intersected with the vertical line of the 'graph', using those coordinates as the axis, I used the compass, which remained at the original setting; to inscribe two more circles. Then, starting at the point(s) of intersection, of the original circle, on the vertical; which are two in number; I used the intersection(s);which were formed of the intersections of original circle and of the two separate circles on the vertical axis; these comprised four additional points. Then, when I connected all these points; from point to point; I found that I had drawn a regular hexagon. Which has six equal sides, and six equal angles; and which; the six points fell on the circumference of the circle I used to define the coordinates, thereof. as I used a compass, that was used--unchanged--in the plotting, or graphing; the coordinate points in the two-dimensional--pencil and paper--diagram; I thus perforce proclaim that the correlation of the radius to the line of circumference, is 1 to 6. Okay so from a quick trip to see if this is actually a hexagon, I find this hexagon was described by Euclid and is one of his 'elements' and appears in book four, proposition 15. Doesn't appear to have a name although; as in Euclid's Hexagon.

Since you are here to ask a question, your primary task is to phrase the question as simply and clearly as you can. You get no "style points" for florid descriptions if their purpose is only to make you appear (to some) better versed in arcane sentence construction, but make it more difficult for the reader to understand what you're asking. The part above is fine, but what you have below is better suited for posting on a site dedicated to English as used in the Elizabethan period.

And since, therefore, as Simon did not understand the relation, I do so pronounce that perhaps Euclid did not neither, therefore; I dost pronounce that it shall be known as McMahan's variant to Euclid's 15 proposition, and dost also challenge that personage to defend his proposition; or explain his propositions in a more formidable manner. On guard Euclid, on guard thou noble geomatarian. Heh, spell check wanted me to replace that with vegetarian, can you imagine.

Yes I can. I doubt that "geometarian" is a word. Someone who is skilled in geometry is usually referred to as a geometer.

I don't have anything against the use of flowery language or sesquipedalian words. However, if their use makes it difficult for a potential responder to parse what you're saying, it behooves the asker to choose clarity over floridity.

 

[Mark44]

Thread moved to General Math, as it had nothing to do with Differential Geometry.

 

[jeffrey c mc.]

I can see the lines for how you constructed the picture. If you'd just filled the page with intersecting circles like the seven you drew, and connected the dots in diamonds, it would be an isometric graph.

[Simon there are only five circles. The center one, and the four around the center circle.

I'm sorry, I don't see why that follows "perforce" from what you drew.
Where did you get that 1:6 ratio from?

One radius, and, six equal divisions of the circumference. The relation is not that they are numerically equivalent; as in having the same measure; but each 'chord' which has as its' base, the straight lines of the hexagon, is related to the radius; as the radius, is equivalent with the geometric center of the hexagon, and the vertices of the hexagon; by which, all (vertices) share the same coordinate point lying on the line of the circumference, thus, one radius, six equal divisions, regardless of whether they are the straight lines of the hexagon, or the 'chord of the circumference. I'm quite gratified that sentence was extraordinarily long and used several semi-colons.

If the distance from the center to a corner of the hexagon is 1 unit, then the perimeter is 6 units - this what you mean? But if I set 2 units to be the perpendicular separation of opposite sides I get a different ratio.

No that was not what I meant. See above

It's unclear what you mean by "correlation" too - you really have to stop using technical words in a poetic way when you are trying to make yourself understood. You don't really mean to say that the ratio of the circumference to the radius is 6 do you?

Yes, in a bold new way, a way no man has gone before.

Also - considering it is you who is making the odd claim, it is you who must elucidate and defend it - which is handy since Euclid is unavailable right now ;)

I rise to that challenge and within three days shall provide you with an alternate equation for solving for the circumference of a circle. As I have already done so, less then 5 years hence; as I already related in another post; all I have to do is review the general knowledge, reapply it; and then perhaps I'll get a little more respect around here. By demonstrating I am not, in fact, a dolt.

 

[Mark44]

I rise to that challenge and within three days shall provide you with an alternate equation for solving for the circumference of a circle.

It's well known (for over 2000 years) that a circle of radius r has a circumference of C = 2##\pi##r.

As I have already done so, less then 5 years hence;

Less than five years from now? That's what "hence" means, the opposite of "ago."

as I already related in another post; all I have to do is review the general knowledge, reapply it; and then perhaps I'll get a little more respect around here. By demonstrating I am not, in fact, a dolt.

 

[jeffrey c mc.]

Mark 44;

Well thank you for humbling me with my failure to write the appropriate words correctly. Well then, since I have a hard time considering hence to refer to the future, I assumed that if I did something in the past I could consider it to be hence. With that said then, if I did something less then 5 years ago, I could relate that it was less then 5 years hence. When you want to tell about a 'sequence of events' or are giving instructions in a step-by-step order, the word then is necessary. If I was comparing 5 to another specific; as five is less than four, then than is correct. Thanks for the opportunity to learn more.

2000 years ehh, no wonder Jesus is so famous. Yet, I will tell you that when you take the Radius, multiply it by the ??Tangent?? of a 30 degree angle, all that is needed is a number expressed as a percentage, to solve for the circumference. The reason for the ??bracket?? is that I did this less then 5 years hence; which should indicate, less then 5 years ago. Now you got me wondering. If less then, is less than something, is something more than less, if so then, then something is more than less, then. When I have done something less then five years hence, I have not done it more than five years ago. One is a sequence, the other is a comparative. One falls within the time allotted, the other asks one to think of the five years, and that which lies beyond, two separate entities. When I say less then 5 years ago, I am only speaking of the time in which I did it, and that it is less than 5 years, except in this sentence [second usage] in which it is used as a comparative; the time I did it, and the 5 year limit. Enough long sentences with two many commas and semi-colons?