Was there another way of describing a circular object?

[Robittybob1]

It came up in another thread where this ancient text did not describe Pi adequately, but in my research on how to calculate Pi I could see there was an alternative way of describing a circle, and most of us would have done this where the circumference can be marked divided by the radius 6 equal times using a compass.

Steps:

Using the compass to draw the circle.

Using the same compass place the point on the circumference and draw a section onto the circle.

move the point to where the section intersects the circle and repeat 6 times.

There will be 6 points on the circle evenly spaced by the radius.

Isn't this what is being described here?

 

 

1 Kings 7:23King James Version (KJV)

23 And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.

 

Some translate "did compass it round about" that as "circumference" but could it be describing the use of a compass to go around it? Then it would be correct as per the diagram. http://sveta-geometrija.com/wp-content/uploads/2011/05/1616.jpg

 

The straight line distance between each point on the circle is equal to the radius (6 times 5 = 30).

Edited June 2, 2016 by Robittybob1

[John Cuthber]

Another pointless exercise would be to calculate the size of a planet here the distance (measured on the bulging surface of the sea) would be 10 units and the circumference was 30 units.

Edited June 2, 2016 by John Cuthber

[Robittybob1]

Another pointless exercise would be to calculate the size of a planet here the distance (measured on the bulging surface of the sea) would be 10 units and the circumference was 30 units.

I don't understand that at all sorry.

If I finally understand you correctly, the "regular hexagon" method of measuring a circle would not be useful in all situations, as in your planet example. But at the beginning measurements were made using dimensions found on the body e.g "a foot 12 inches, a yard the length of a stride, for there was no standardised rulers etc.

Edited June 2, 2016 by Robittybob1

[Strange]

It came up in another thread where this ancient text did not describe Pi adequately, but in my research on how to calculate Pi I could see there was an alternative way of describing a circle, and most of us would have done this where the circumference can be marked divided by the radius 6 equal times using a compass.

Steps:

Using the compass to draw the circle.

Using the same compass place the point on the circumference and draw a section onto the circle.

move the point to where the section intersects the circle and repeat 6 times.

There will be 6 points on the circle evenly spaced by the radius.

Isn't this what is being described here?

 

Some translate "did compass it round about" that as "circumference" but could it be describing the use of a compass to go around it? Then it would be correct as per the diagram. http://sveta-geometrija.com/wp-content/uploads/2011/05/1616.jpg

 

The straight line distance between each point on the circle is equal to the radius (6 times 5 = 30).

 

 

You are describing a hexagon, not a circle.

[Robittybob1]

 

 

You are describing a hexagon, not a circle.

Yes, the hexagon that fits exactly in the circle, one defines the size of the other. The length of the side of the regular hexagon is exactly the radius of the circle. It is a very special case.

Edited June 2, 2016 by Robittybob1

[Strange]

It may be a special case, but it isn't a circle or "round".

[Robittybob1]

It may be a special case, but it isn't a circle or "round".

You might have to explain that a bit more. Are you using "round" as in the original quote?

"it was round all about"

 

I take that to mean it was hemispherical.

[Prometheus]

Yes, the hexagon that fits exactly in the circle, one defines the size of the other. The length of the side of the regular hexagon is exactly the radius of the circle. It is a very special case.

 

Are you talking about the method of exhaustion Archimedes used to find bounds on pi, whereby a circle is approximated by a polygon - the more sides the polygon has the more narrow are the bounds (how close were the Greeks to understanding limits and maybe then calculus?).

 

Apparently the method was first developed around the same time that the old testament was written so it's possible, but we are unlikely to ever actually know. If it can, somehow, be shown that this is the case it would be a nice little piece of evidence that the old testament was written only by humans.

[Strange]

You might have to explain that a bit more. Are you using "round" as in the original quote?

I take that to mean it was hemispherical.

 

 

So, not a hexagon.

[Robittybob1]

 

 

So, not a hexagon.

Do you understand what I was suggesting using the regular hexagon for? You can define any circle by the max. sized regular hexagon that fits exactly within it. The points of the hexagon must share 6 points with the circle.

http://www.mathopenref.com/constinhexagon.html

https://en.wikipedia.org/wiki/Circumscribed_circle explains the special nature of the circumscribed hexagon.

Here is a historical based reference: http://www.geom.uiuc.edu/~huberty/math5337/groupe/overview.html

 

The ancient Babylonians knew of the existence of pi - the ratio of the circumference to the diameter of any circle. The constant they obtained, 3.125, made use of their knowledge that the perimeter of a regular hexagon inscribed in a circle equals six times the radius of the circle. By using this perimeter of the inscribed hexagon as a lower bound for the circumference of the circle, they were able to come up with their remarkably close approximation for pi circa 2000 B.C. [2, p.21].

Edited June 2, 2016 by Robittybob1

[imatfaal]

You can uniquely define a circle by the edge-length and number of edges of any regular polygon; ie only one circle will touch all the vertices. The clever bit is that you can also define a unique polygon which fits outside the circle - with the centre of each edge touching the circle. You can thus place a strict constraint on the circumference of a circle - ie it has to be greater than the sum of the sides of the polygon within the circle and it has to be less than the sum of the sides of the polygon outside the circle. As a regular n-polygon can be deconstructed into n isosceles triangles (with two edges the same length as the radius of the circle for the inner polygon OR a perpendicular height equal to the radius for the outer polygon) the edge-length can be calculated exactly. Thus with just a pen and paper and a huge amount of patience pi can be estimated to any arbitrary accuracy

[Strange]

Do you understand what I was suggesting using the regular hexagon for? You can define any circle by the max. sized regular hexagon that fits exactly within it. The points of the hexagon must share 6 points with the circle.

http://www.mathopenref.com/constinhexagon.html

https://en.wikipedia.org/wiki/Circumscribed_circle explains the special nature of the circumscribed hexagon.

Here is a historical based reference: http://www.geom.uiuc.edu/~huberty/math5337/groupe/overview.html

 

That has already been said (including in the other thread). Which is part of the reason why the Bibles statement that pi=3 is especially nonsensical. There was better information available. (I suppose that story could be even older than Babylonian mathematics.)

[imatfaal]

More interesting - to me at least - is how the ancients worked out the edge length (as a relation to the given side obviously) given the internal angle and one side. I can do it with trigonometry - but I wonder if they had geometric rules and tables that were worked out in another method.

 

For instance - a six sided polygon is made up of equilateral triangle and getting the edge lengths is easy, I could probably work out a few others with nice divisors; but from memory greek scholars were using 100+ sided shapes to work out pi constraints

[Robittybob1]

 

That has already been said (including in the other thread). Which is part of the reason why the Bibles statement that pi=3 is especially nonsensical. There was better information available. (I suppose that story could be even older than Babylonian mathematics.)

The way I was suggesting they don't need to know pi, they just measure the total length of the sides of the inscribing polygon (hexagon) which is in this case 6 times radius a process I've provisionally called "compassing" (for I don't know what else to call it) but from there the words describing that action are derived e.g encompass. Google definition.

verb

1.

surround and have or hold within..

 

Even the concept of the "radius" rather than just the diameter had to be developed.

Today we would describe a hemisphere by its radius and all the rest can be worked out from that.

[John Cuthber]

I don't understand that at all sorry.

If I finally understand you correctly, the "regular hexagon" method of measuring a circle would not be useful in all situations, as in your planet example. But at the beginning measurements were made using dimensions found on the body e.g "a foot 12 inches, a yard the length of a stride, for there was no standardised rulers etc.

It's nothing to do with the units.

Imagine a planet where one hemisphere is sea and the other is land.

To go round the sea you have to sail round the circumference.

To sail in a "straight" line across the sea from one point on the coast to the furthest point you have to sail half the circumference (taking a shorter route would need a submarine).

So, for this sea the ratio of the circumference to the diameter is 2.

In that world, "pi" is two for a big enough circle.

For a small circle the sea is approximately flat so "pi" would be 3.14159.....

Somewhere in between, there is a size of circular sea where the "diameter" is exactly a third of the circumference.

 

If you wanted to, you could calculate the size of the sea that meets this criterion (in terms of the radius of the planet).

 

And it would still be just as pointless as pretending that a hexagon is circular or that pi is 3..

Edited June 2, 2016 by John Cuthber

[Robittybob1]

It's nothing to do with the units.

Imagine a planet where one hemisphere is sea and the other is land.

To go round the sea you have to sail round the circumference.

To sail in a "straight" line across the sea from one point on the coast to the furthest point you have to sail half the circumference (taking a shorter route would need a submarine).

So, for this sea the ratio of the circumference to the diameter is 2.

In that world, "pi" is two for a big enough circle.

For a small circle the sea is approximately flat so "pi" would be 3.14159.....

Somewhere in between, there is a size of circular sea where the "diameter" is exactly a third of the circumference.

 

If you wanted to, you could calculate the size of the sea that meets this criterion (in terms of the radius of the planet).

 

And it would still be just as pointless as pretending that a hexagon is circular or that pi is 3..

To think in these terms is very different than anything I've struck before. I'm not the quickest on uptake but if someone else could comment I might get it.

I have not consciously said "that a hexagon is circular or that pi is 3". A regular hexagon inscribes a circle. That hexagon has sides equal to the radius of the circle. That is what I have said.

 

Note "inscribes" is not the same word as "describes" as used in the title of the thread.

Edited June 2, 2016 by Robittybob1

[John Cuthber]

The hexagon does not go round the circle- it goes through it and crops bits off.

To go round it you would need to use the hexagon that circumscribes a circle.

That's rather bigger

 

http://geometryatlas.com/entries/190

[Robittybob1]

The hexagon does not go round the circle- it goes through it and crops bits off.

To go round it you would need to use the hexagon that circumscribes a circle.

That's rather bigger

 

http://geometryatlas.com/entries/190

The title of that page is a clue: "Inscribed and Circumscribed Circles of a Regular Hexagon". To you the circle seems to be the area within it, but to me the circle is just a special line.

You could be right, maybe I should say "boundary of the circle" e.g "draw a circle" should be "draw the boundary of a circle". Interesting.

Edited June 2, 2016 by Robittybob1

[Strange]

To think in these terms is very different than anything I've struck before. I'm not the quickest on uptake but if someone else could comment I might get it.

 

 

This is an example of non-Euclidean geometry.

https://en.wikipedia.org/wiki/Non-Euclidean_geometry

[Robittybob1]

 

 

This is an example of non-Euclidean geometry.

https://en.wikipedia.org/wiki/Non-Euclidean_geometry

Up till now whenever I see the word "Non-Euclidean" I switch off. I have not even attempted to master that yet. Have you understood it fully?

[Capiert]

Up till now whenever I see the word "Non-Euclidean" I switch off. I have not even attempted to master that yet. Have you understood it fully?

Hi, I'm fascinated with your observation, even if the rest aren't.

Euclidean geometry is straight line geometry.

So non-straight geometry is curved.

It doesn't have to be an exact circle, either.

They** (=the math world) make the distinction, because straight line is easy y=m*x+b, known.

(It's linear, x is the culprit, with a factor m which makes the slope (affecting the angle);

& hang_on "+b" which is the offset (raising it) up the y_axis.)

But things can get a bit squirrelly when they start to curve.

A circle is relatively easy (=known, uses rooting which begins to get complicated), but things get more complicated when they are not exactly obeying pythagorus r^2=x^2+y^2.

In fact very many curves have nothing to do with circular geometry.

Best wishes to all.

 

Theme: (non)Euclidean.

 

** If I said we, it wouldn't be in laymans terms. Informal to get the idea across.

 

Euclid was a greek & invented it (straight line (=Euclidean) geometry) with only sentences

(observations written down, now known as laws or axioms, something like that.)

So they named it (his work) after him.

 

non_Euclidean: It's everything else, which is not straight (no exceptions).

Edited June 3, 2016 by Capiert

[Robittybob1]

One day I'll pluck up courage and look at non-Euclidian geometry.

[Capiert]

One day I'll pluck up courage and look at non-Euclidian geometry.

Enjoy it!

 

Einstein said:

Work+Fun=Success.

 

Cheers.