Coal and Math

CombatWombat51

First Post
I'm doing some mapping stuff for my campaign, and I have two seperate questions that I figure some kind souls here could help me with :D

1. What was coal used for before steam power? Was it heavily mined?

2. How can one calculate the area of a hexigon?
 

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If the hexagon is regular (all angles the same and all sides of equal length), square the length of a side and multiply by 3/2*sqr(3), which is about 2.598. The area of a hexagon with a side 3cm is 3 * 3 * 2.598 ~= 23.38 cm^2.

If not... there are other ways, if you'd describe what you know about the hexagon.
 

CRGreathouse said:
If the hexagon is regular (all angles the same and all sides of equal length), square the length of a side and multiply by 3/2*sqr(3), which is about 2.598. The area of a hexagon with a side 3cm is 3 * 3 * 2.598 ~= 23.38 cm^2.

If not... there are other ways, if you'd describe what you know about the hexagon.

As for what I meant, I meant the regular hexagon you described. 6 sides of equal length and equal angles.

So let me see if my rusty algebra pulls through, using a hexagon with a side 5 miles long. I'll use 2.5 instead of 2.598 since this doesn't need to be completely exact, just close enough.

5 miles * 3 * 2.5 = 37.5 mi^2?

Thanks for the help, CRG.
 

Hexagon - more easily it's (1 side squared) * 2.6 = rough estimate of area. If you need precision, follow the guideline above.

As for coal - Interestingly enough it was used quite often long before steam. Egyptians knew of it and used it for religious activities as well as a source of heat. Greeks knew of it - look up someone named Mechanus, he actually invented a steam toy ~1000 BCE :)

Generally speaking, most cultures knew of charcoal and coal. The indians (of india) used it for smithing purposes long before most other cultures. Ever hear of Damascene steel? It's a forgery (pun intended) but if you research it, you'll hit the origins of folded steel - hotly contested topic, but essentially folded steel goes WAY back in time like to near end of bronze age according to some (though I don't believe it) and air-fueled COAL fires are the only thing hot enough to create the right mix of alloys for it.

Coal was often used in a variety of civs for incense burning and/or religious purposes.

As for whether it was heavily mined - well, that depends on era and time. The early Harappan civilization of the Indus continent at ~2500 BC extensively mined the stuff - but no other culture of the time did. The egyptians and the greaks of 2000 to 500 BC used it, so I assume they did some extent of mining for it. Romans used it for coal-fired hot public baths and steam rooms - not to mention torture, so I'd assume they had a source for it as well.

If you need more details/references lemme know - I'm an internet researcher by trade :)
 

For the hexagon, unless I'm reading CRG's wrong, both those are equations give pretty different results. His is side^2 * 3 * 2.5 = area. Yours is side^2 * 2.6 = area. So am I reading an extra * 3 in CRG's, or am I missing a * 3 in yours? :confused:

Thanks for that info on Coal, Tilla. I was trying to decide if coal was valuable enough to consider when figuring significant natural resources of an area, and I was wavering. Now I'm not :D
 

your formula for finding the area of regular polygons -- A = (0.5p)(0.5p)/{n[tan (180/n)]} -- is correct. Here is the proof.



We already know the formula for the area of a regular polygon is...

(1) A = 0.5ap, where a is the apothem and p is the perimeter.

We also can find another relation for a regular polygon by dividing the polygon into n isosceles triangles using the radii and dividing each of those triangles into 2n right triangles with the use of the apothems. The formula is...

tan (360/2n) = 0.5s/a, where s is the length of each side of the polygon. The formula can be cleaned up to...

(2) tan (180/n) = 0.5s/a.

We can solve relation #2 for a and get...

(3) a = 0.5s/[tan (180/n)]

Substitute relation #3 into relation #1 and get...

A = 0.5{0.5s/[tan (180/n)]}p.

Let's multiply the right side by n/n to get...

A = 0.5n{0.5s/[n[tan (180/n)]]}p. Let's rearrange the terms to get...

A = 0.5ns{0.5p/[n[tan (180/n)]]}. Keep in mind that p = ns; the perimeter is equal to the length of one side times the number of sides. Let's make this substitution into the last relation for area to get...

A = 0.5p{0.5p/[n[tan (180/n)]]}. If we regroup, we can get... A = (0.5p)(0.5p)/{n[tan (180/n)]}. This was your proposed formula for area; therefore, the formula you gathered by using pattern-recognition as a guide is absolutely correct.


OR Go to this link. http://www.drking.plus.com/hexagons/misc/area.html

Aluvial
 

And the award for the weirdest subject line on a message board goes to....


*opens envelope*

CombatWombat51, for his elusive, intriguing "Coal and Math".
 

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CombatWombat51 said:
As for what I meant, I meant the regular hexagon you described. 6 sides of equal length and equal angles.

So let me see if my rusty algebra pulls through, using a hexagon with a side 5 miles long. I'll use 2.5 instead of 2.598 since this doesn't need to be completely exact, just close enough.

5 miles * 3 * 2.5 = 37.5 mi^2?

Thanks for the help, CRG.

His original thing wasn't 3 cm * 3 * 2.598, it was 3cm * 3cm *2.598. You left in that second factor of three instead of making it be 5^2.

As was noted elsewhere, the proper formula is (about) 2.6*radius^2. This should be more than accurate enough for you.
 

I'm stating the obvious, but coal is a fuel-source.

Industries that require constant fires (foundries, smithies, breweries, etc.) may very well use coal instead of wood if its available, especially if trees are scarce in the area. Coal also burns hotter and longer than wood, increasing its appeal.
 

CombatWombat51 said:
1. What was coal used for before steam power? Was it heavily mined?

You are going to have to be careful about this, because a lot of what was called 'coal' in early records was charcoal (made by charring wood), not mineral coal. For example, when you read about a saint being martyred by burning 'over coals', they mean that he was barbecued over the coals of a wood fire, not that he was burned in a coal fire. And when you read of somebody being tortured with hot coals, those coals will be burning embers taken from among the coals of a wood or charcoal fire. If you come across a reference to Egyptians, Greeks, or Romans using 'coal', be sure that it mentions that they mined it, because maybe "carbo" has been translated as "coal" where "charcoal" or even "ember" would have been better. Consult decent dictionary, and you will see that the primary sense of the word 'coal' is "a piece of glowing carbon or charred wood: ember", the secondary sense is "charcoal", and the usual modern sense is only tertiary. This sort of thing misleads the people who compile translating dictionaries.

Marco Polo describes mineral coal as one of the wonders of his stay in China: a black stone that burns better than wood etc. So evidently it was unfamiliar in Venice in his time.

But by the Late Middle Ages (Wikipedia says 'after about 1000 AD') "sea coal" as distinct from "char coal" was an article of commerce in England and Scotland (and probably other northern countries that I don't know of). It was mostly burned in fireplaces to heat rooms, because food cooked over an open coal fire picks up an unpleasant taste (and traces of some pretty nasty organic poisons), and coal as such is of little use in metallurgy. Metallurgists and smiths invariably used charcoal in their industry, not coal, at least until coke (which you can think of ascharcoal made out of coal rather than wood) was invented in the 17th Century. If you come across any reference to 'coal' used in making steel, there is something wrong (probably mistranslation, or else somebody believing a dictionary that tells him that the result of charring wood is called 'coal' in English).

Coal was mined on [what we would now call] a small scale in the North of England in late mediaeval times, and shipped south in specially-built coaling ships.

As for Tilla the Hun's "Mechanus", you will search in vain. He seems to be thinking of the aeropile, which was invented (along with a number of other amazing tricks such as the first known coin-fed vending machine) by Hero of Alexandria. (Also sometimes transliterated as 'Heron' rather than 'Hero'.) Many of Hero's gadgets were powered by the heat of fires, but those could easily have been wood or charcoal fires. If you find a source that seems to say that they used coal fires, be careful about the translations. Make sure that the original source mentions mining or a burning stone.
 

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