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Old 09-27-2018, 08:37 PM   #1
hal
 
Join Date: Aug 2004
Location: Buffalo, New York
Default Integrating T5 material with GURPS TRAVELLER

Hello Folks,
I'm trying to use material from GURPS with GURPS TRAVELLER and using some of the material from Traveller 5th edition for background material. To wit:

The area of a Fief Hex.

I'm running into a problem however, and I can't help but think fault is with me instead of with T5. On the off chance that I am correct in this, then there is a problem with T5 that will need an errata issued on it.

A while back, I came across a quick and dirty way to estimate the area of a hex. It was simple and to the point:

Simply put, if you measured from side to side of a hex (ie, a line drawn from the center of one side, through the center of the hexagon, and to the other side at the center of the line), you could estimate the area of the hex as follows:

Find the width from face to face, and multiply that by 0.866, then multiply the two together to get your area.

For example, if your hex is nominally 25 miles wide from face to face, its area will be 25 * (25 * 0.86 = 21.5) = 537.5 square miles.

But T5 stipulates the following from various pages spread throughout the book:


Traveller 5th edition says this on page 421:

"Hex Size. Hex size (or hex diameter) reflects the distance from the center of a hex to the center of an adjacent similarly sized hex. Hexes are universally even decimal multiples of meters (100 meters, 1,000 meters, 10 kilometers, and 1 km)."

This is the way to determine what the Apothum is - which would be half the distance between center of hex to center of hex (or the line that separates them from each other).

Page 459 of Traveller 5th edition has this to say about local hex:

"Mapped In Hexes. The Traveller Mapping System defines a hierarchy of mapping hexes: the 1000 km World Hex; the 100 km Terrain Hex; the 10 km Local Hex; and the 1 km Single Hex."

Page 693 of Traveller 5th edition has this to say about fief hexes as local hexes:

"Outright Ownership of one Local Hex (approximately 65 square km= 6500 hectares= 16,000 acres)."

Doing the math? A 10 km hex radius implies a 10 km side to side distance within a hex. It also implies an Apothem of 5 km. When my quick and dirty gave me an answer of 8660.25 Hectares, or 21,399.95 acres, I decided to dig deeper and get an actual formula that wasn't quick and dirty. When the actual formula matched the quick and dirty method - I figured "ok, either the math is off, or there is an Error in T5."

Can anyone confirm what the area of a 10 Km side to side hex should be?
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Old 09-28-2018, 08:41 AM   #2
Fred Brackin
 
Join Date: Aug 2007
Default Re: Integrating T5 material with GURPS TRAVELLER

Quote:
Originally Posted by hal View Post
Can anyone confirm what the area of a 10 Km side to side hex should be?
My instinct is to treat a hexagon as if it were a circle where Pi is equal to 3.

Besides being part of an old Gurps joke it does seem obvious to me that a hexagon is smaller in area than a circle of equal diameter but not hugely smaller.

The area of a circle is Pi x R squared and the area of a circle 10 km in diameter simplifies to 3.14 x 5 km squared or approximately 248 square kilometers.

3x5 sqaured is much simpler and comes to exactly 225 aquare kilometers.
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Old 09-28-2018, 10:13 AM   #3
swordtart
 
Join Date: Jun 2008
Default Re: Integrating T5 material with GURPS TRAVELLER

3 x 5 x 5 = 75.
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Old 09-28-2018, 10:46 AM   #4
hal
 
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Location: Buffalo, New York
Default Re: Integrating T5 material with GURPS TRAVELLER

Quote:
Originally Posted by Fred Brackin View Post
My instinct is to treat a hexagon as if it were a circle where Pi is equal to 3.

Besides being part of an old Gurps joke it does seem obvious to me that a hexagon is smaller in area than a circle of equal diameter but not hugely smaller.

The area of a circle is Pi x R squared and the area of a circle 10 km in diameter simplifies to 3.14 x 5 km squared or approximately 248 square kilometers.
I'm not throwing stones here, as my own math at times can slip up only too easily... *teasing grin*

3.14 x 25 = 78.5

Dividing 248 by 3.14, I get 78.9. My guess is that if you used a calculator, you may have hit the enter button twice instead of once... (just a guess)

So, 78.9 sq km is close to 86. I think the only real difference is that whereas I used the value of the radius as 5.773502692 km instead of 5, and used the value of pi with as many decimals as Excel would permit. This "ballpark value" ended up being about 104.7197551 sq km.

Below are a few more "attempts" showing all work etc (inclusive of your ballpark estimate idea). Skip if you're not interested, but it does show alternate ways of arriving at what appears to be the same answer.

Method 1: Rough rule of thumb where Width side to side x a constant equals area

Method 2: Formula for calculating a hexagon based on the Apothem and the length of a side of the hexagon (incidentally, also the length of a line from the center of a hexagon to its junction between two sides)

Method 3: Formula for area of a triangle knowing height of triangle (Apothem) times base of triangle (hexagon side length)/2 and then multiplying it all by 6

Method 4: Determine the area of a rectangle whose length is the side to side measurement and whose other dimension is its Apothem. Then determining the area of the two remaining triangles inside the hexagon, but external to the rectangle inscribed within.

Method 5: determine the area of a circle whose radius is equal to the side of a hexagon. Multiplying this by .827 (82.7%) seems to be in the same ballpark as methods 1 through 4.


What follows are the three additional methods not in the original first post...

We know that the apothem height is 1/2 Width of the hex itself from side to side. We know that the formula for the length of a hexagon side is 2 x apothem x tan(30 degrees), we get a result that the length of a side of the hexagon is 5.773502692 where the width of the hexagon is 10 km.

Area of a triangle is 1/2 base x height. So we have a height of 5 km, and a base of 5.773502692 km. Area is 14.43375673 sq km. Multiply that by 6 for the number of equilateral triangles, and we get 86.06254 sq km.

Trying a different route...

If you inscribe a rectangle within the Hexagon, its sides would be 10 km x 5.773502692 km. That's 57.73502692 for the rectangle alone. We still have two triangles remaining. So, taking ONE of those triangles, we know that its base is 10 km. We know that one of the sides is 5.773502692. The line drawn from the vertex of the triangle will bisect it in half, which is 60 degrees. So that makes it a 30 degree angle. The line drawn from the vertex to the base is going to be equal to tan(30) degrees * 1/2 base length (As an angle bisector will also cut the line in half that it is perpendicular to. This gives us 2.886751346 km. So, area of a triangle is 1/2 base times height, but we have two such triangles, the area becomes 10 km * 2.886751346 or 28.86751346 sq km. That plus 57.73502692 = 86.06254 sq km.

So now I've gone four routes in trying to calculate the area of a Hexagon. The quickie rule of thumb, the actual formulas for deriving Side length of a hexagon based on apothem length, as well as using the Perimeter formula involving both Apothem and side length. Last but not least, using the area of a rectangle plus the two remaining triangles left after removing the area of the rectangle from the hexagon figure.

If the "Radius" formed from the center of a Hexagon drawn to junction of the angle formed by two sides is the radius of a circumscribed circle - then, Radius equals the value of a side (as a hexagon is formed by perfect equilateral triangles formed with the same length per side). If the radius is equal to 5.773502692, then the area of a circle is r^2 x pi. This works out to....

104.7197551 sq km

This implies that roughly 17.3% of the circle is external to the area external to the hexagon, but internal to the circle. Hmm. I wonder if this relationship holds true for all perfect hexagons inscribed in a circle (ie, the area of a hexagon is equal to approximately 82.7 % of the area of a circle). Would be interesting if true.
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Old 09-28-2018, 10:47 AM   #5
hal
 
Join Date: Aug 2004
Location: Buffalo, New York
Default Re: Integrating T5 material with GURPS TRAVELLER

Quote:
Originally Posted by swordtart View Post
3 x 5 x 5 = 75.
Ninja'd!

Nice eyes there...

;)
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Old 09-28-2018, 07:29 PM   #6
Fred Brackin
 
Join Date: Aug 2007
Default Re: Integrating T5 material with GURPS TRAVELLER

Quote:
Originally Posted by hal View Post
Ninja'd!

Nice eyes there...

;)
We've got a parenthesis issue and obviously mine. I did not hit "enter" twice. I mulitplied 3x5 and squared that. 15 squared is indeed 225 just as 5 squared x 3 is 75.

It was bothering me that a square that was 10 kn per sidewas definitely 100 sq. km while i was getting bigger numbers for a hex with that as its' longest measurement. Obviously the hex fits inside the square and must be smaller in area.

I'm sorry i can't comment on your other methods but they were just tl:dr. :)

I should know better than to answer math questions but the answer apperared so simple to me and once I straightend out my parentheses it was.
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Old 10-01-2018, 05:58 PM   #7
Anaraxes
 
Join Date: Sep 2007
Default Re: Integrating T5 material with GURPS TRAVELLER

Given the length of a side:
a = 3 sqrt(3) / 2 * l^2

The length of a side isn't usually given, alas.

So, the sides of a 30-60-90 triangle are in the ratio 1 : sqrt(3) : 2. So, your apothem = sqrt(3) / 2 * l. The apothem is half the side-to-side distance, or the center-to-center distance. (Which is usually what people mean when they say something like "a 10 km hex"; moving from one hex to the next takes you 10 km.) If we call that distance "scale", then

scale / 2 = apothem = sqrt(3) / 2 * l.

scale / 2 * 2 / sqrt(3) = l = scale / sqrt(3).

Third derivation, find the length of the hex side in terms of scale, and plug that into that first textbook formula.

l = hscale / sqrt(3) * 2 = scale / sqrt(3).


Back to the first formula:

area = 3 sqrt (3) / 2 * (scale / sqrt(3))^2
area = 3 sqrt (3) / 2 * scale ^2 / 3

area = sqrt(3) / 2 * scale^2

86.6 km^2 = area of 10 km hex

Or for another derivation, you could calculate the area of one of the little triangles, center to midpoint of a side (half the scale), and multiply by 12 (two little triangles per side of the hex). I'll call half the scale "hscale", just because it's easier to work with. The area of the little triangle is half the area of the rectangle formed by the two shortest sides. But then, we want 12 of those anyway, so we can just take 6 of those rectangles.

area = 6 * hscale * hscale / sqrt(3) = 6 * 5 * 5 / sqrt(3) = 86.6 km^2

Last edited by Anaraxes; 10-01-2018 at 06:03 PM.
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Old 10-01-2018, 06:36 PM   #8
hal
 
Join Date: Aug 2004
Location: Buffalo, New York
Default Re: Integrating T5 material with GURPS TRAVELLER

Quote:
Originally Posted by Anaraxes View Post
Given the length of a side:
a = 3 sqrt(3) / 2 * l^2

The length of a side isn't usually given, alas.

So, the sides of a 30-60-90 triangle are in the ratio 1 : sqrt(3) : 2. So, your apothem = sqrt(3) / 2 * l. The apothem is half the side-to-side distance, or the center-to-center distance. (Which is usually what people mean when they say something like "a 10 km hex"; moving from one hex to the next takes you 10 km.) If we call that distance "scale", then

scale / 2 = apothem = sqrt(3) / 2 * l.

scale / 2 * 2 / sqrt(3) = l = scale / sqrt(3).

Third derivation, find the length of the hex side in terms of scale, and plug that into that first textbook formula.

l = hscale / sqrt(3) * 2 = scale / sqrt(3).


Back to the first formula:

area = 3 sqrt (3) / 2 * (scale / sqrt(3))^2
area = 3 sqrt (3) / 2 * scale ^2 / 3

area = sqrt(3) / 2 * scale^2

86.6 km^2 = area of 10 km hex

Or for another derivation, you could calculate the area of one of the little triangles, center to midpoint of a side (half the scale), and multiply by 12 (two little triangles per side of the hex). I'll call half the scale "hscale", just because it's easier to work with. The area of the little triangle is half the area of the rectangle formed by the two shortest sides. But then, we want 12 of those anyway, so we can just take 6 of those rectangles.

area = 6 * hscale * hscale / sqrt(3) = 6 * 5 * 5 / sqrt(3) = 86.6 km^2
Thanks Anaraxes.

Posted on another thread in another forum, this is what I wrote...

"So why all this attention to detail? Per Marc Miller, each "Local Hex" is a 10 km hex, which makes it a 5 km Apothem, which in turn makes the area of a 10 km hex equal to 86.60254038 square km, or 8660.254038 hectares, or 21,399.9537766255 acres - not the approximately 65 sq km or 16,000 acres."

86.60254 sq km is about 21,399.95378 acres.

Thanks again for the confirmation that I wasn't imagining things. :)
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