(TFT) Moria Maps. Hex Geometry & Scaling Factors of 2.645 --> TFT wargame

[Rick Smith <rsmith@lightspeed.ca>]

// There seem to be a variety of "best of" posts 
// lately on the TFT mailing list.   Here is one
// of mine, (with significant edits) with a big 
// bunch of new content at the bottom.


Hi Dave,
  You could pick up a copy of Lords of the Underearth
(LotU) but it would give you a pre-made (tho very 
small) Dwarven mine.  (I would change residential 
areas shown by a single room into clusters of 
apartments, when turning it into an adventure.)

  Thinking about the size.  The hexes in LotU are
Mega-Mega-Mega hexes.  (I'll call this a Level 4 hex.) 
Such hexes are <36 meters across, and have an area of 
exactly 343 hexes  (a single hex is ~1.59600 meters 
square if my trig is correct so a Level 4 hex would 
have an area of 528.1 square meters).

  (5.28 hectares should be able to feed the peasant
farmers on the land plus 1 or 2 more people depending 
on the land quality.)

  Now that distance across I said was <36 meters.  I 
got that number by multiplying 1.333 meters by three, 
3 times.  However, a Mh is not 3 times the width of a 
regular hex.   They pack.   I don't see any easy 
geometric solution, so I am going to measure one of 
my big maps.  

  (Long Pause.)

  Ok, on the Chessex battle map that I've drawn Lv 2
Level 3 and Lv. 4 hexes on, from the center dot to 
center dot of a Lv. 1 hex is 2.51 cm (~one inch).  
This represents 1.333 meters.

  Measuring from the center dot to the center do of a
Level 4 hex, is 46.7 cm across.

  This works out to representing: 24.51 meters across.


  This is pretty close to 5 times the width of a
Mega-Mega hex (Lv. 2 hex).  Notice that in ItL page 
24, second column first paragraph after table at the 
top of the page, Steve Jackson suggests that the size 
of map hexes for larger scale maps be scaled up by 
factors of 5 for later maps.  This would work pretty 
close IF THE ORIGINAL sized hex were 1.0 meters 
across.  Since one hex is 1.333 meters, his scaling 
factor won't match the Mega-hex, Mega-Mega hex growth 
rate.

  Also note that LotU specifically stated that its
hexes were Level 4 TFT hexes.

  Anyway, my calculations show a scaling factor of
2.64.  Multiply 1.333m by 2.64 = 3.5m (the width of 
a Megahex).  Multiply 3.5m by 2.64 = 9.27m (the width 
of a Mega-Mega hex or a level 3 hex).  Multiply 9.27m 
by 2.64 = 24.47m (the width of a Level 4 hex).

  (This is pretty close for a scaling factor of 7
when going up two levels in size rather than SJ's 
scaling factor of 5.  If we want to say that two
levels is exactly 7 then one level is sqrt(7) or
~2.6457513.)  I will use the approximation of 2.645.


  Anyway, back to our original discussion of Moria
and using the LotU map for it.  I've loaned my copy 
away but It seems to me that there were about 22 
hexes across it counting via the hex grain.  (I got
a copy and measured it.  It is 22.5 hexes across.
22 on even # rows and 23 on odd numbed rows.) 

  If we go with the LotU map being 22 hexes across,
the distance thru that mountain is ~550 meters.  Now
Tolkien's mountain ranges (in particular the Misty
Mountains) were unrealistically thin.  But let us say
that this mountain range was from 30 to 50 km wide.  
(I.C.E.'s Moria shows it 40 miles across, so call it
40 km.)  Taking 40 km, Moria should be ~72.7 LotU
maps across.

  (Note Tolkien's mountain ranges were too thin, 
but his individual mountains are too thick.  Most 
mountains are considerably less than 40 km thick.)


Summary:
  To reiterate, the LotU map shows you the doors,
tunnels, and LARGE rooms of a VERY small dwarven mine.
(Small rooms are not shown and must be interpolated.)

  My friend Paul Danielson used to run a TFT campaign
and wanted to do Moria.  He used dozens of pages of 
graph paper (square grid) and mapped Moria like the 
Platt river (a mile wide and an inch deep).  By that 
I mean, that he had a tonne of area mapped, but it 
was just thin lines on a square grid that had no 
character or interesting thought behind it.

  What are peoples feelings about adventuring in a
dwarven palace?  How would you map it?  This is a 
subject of intense interest to me, as I've ALWAYS 
wanted to do an adventure in Moria.


  **********************************************
  // New material:

  One thing that has always been a nuisance in TFT
is the 1 hex = 1.333 meters along the hex grain.
Not only must you multiply distances by 1 & 1/3
but the distances are less than this amount when 
not measured along hex grain.  Let us say, that a
hex is a bit more than a meter and use the 
approximation 1 hex = 1 meter.  When you measure
along the hex grain you under - estimate the 
distance by a bit, but when you count along the 
alternate hex grain you over estimate it by a 
bit.  On average, we can say that 1 hex = 1 meter
and be done with it.

  Using my battle mat:

  10 hexes along hex grain = 25.1 cm.
  10 hexes along alternate hex grain = 19.3 cm.

  The average of this is: 22.2 cm.

  If we say that 22.2 cm (battle map) = 1 meter
(TFT scale) then we find that a hex is:

  1.1306 meters.  (Call a hex 1.13 meters across.)


  ***********************************************

  Note: if we assume that the LotU hexes are 25m 
across then a regular hex is 1.3528m which is too 
big.  At this size we get:

Lv 1 hex:  1.35m
Lv 2 hex:  3.57m
Lv 3 hex:  9.43m
Lv 4 hex: 25m
Lv 5 hex: 66m
Lv 6 hex: 175m
Lv 7 hex: 462m
Lv 8 hex:  1.225 km  ~1.250 km
Lv 9 hex:  3.234 km


  If we assume that the TFT hex is exactly 1.13
meters across (for the reasons given above) then 
the LothU hexes are about 20.9 m across.  Assuming 
a scaling factor of 2.645, then:

Lv 1 hex:   1.13m   ~1 meter   or 13.0 % error
Lv 2 hex:   2.988m  ~3 m        or  4.0 % error
Lv 3 hex:   7.91    ~8 m        or   1.13 % error  
Lv 4 hex:  20.91m  ~21 m        or  0.37 % error
Lv 5 hex:  55.31m   ~55 m       or  0.67 % error
Lv 6 hex: 146.2m   ~150 m       or  2.36 % error
Lv 7 hex: 386m
Lv 8 hex: 1023m    ~1 km        or  2.3 % error
Lv 9 hex:  2.706 km 

  (The % error says how big an error there is
between the actual hex size & the approximation
given after it.)

  These approximations work out quite nicely,
(much better than some I tried).  In particular
the Lv 8 hexes are very close to 1 km across.


  We get about a 10% error if we say that when 
you go up 7 levels of hexes, the hex size 
increases a thousand fold.  We get 2.68 % error 
when we say that Going up 4 levels gives us a 50 
fold increase in size.  So as long as the maps 
use multiple of scales that are 50 and 1000 fold 
bigger than earlier scales, we won't go too far 
wrong.

  This suggests standard TFT maps should be:

Lv  1 hex.  (Exactly 1.13 meters across.)
Lv  2 hex.  (Mega hex.)
Lv  4 hex.  (LotU sized = 21 meters.)
Lv  5 hex.  (My revised LotU ? ~55 meters.)
Lv  6 hex.  (Operational mapping.  150m / hex.)
Lv  8 hex.  (Strategic mapping. 1 km / hex.)
Lv 12 hex.  (Kingdom mapping.  50 km / hex.)
Lv 15 hex.  (Campaign mapping. 1000 km / hex.)

  The level 15 hexes are furthest off, but they
are larger than any fantasy game will likely 
need.  All the other scales here are within a 
couple % points of the listed size.

  Of course other scales could be used, but they 
won't fit into the nice "hexes made out of hexes"
system.  (Actually, how many people are going to
be worried about nesting several levels of maps
inside the hexes of each other?  I've likely 
spent way too much time on this already...)

  ************************************************

  I think I am going to make a standard TFT hex
1.13 meters across (along the hex spine) in my 
campaign.  That would make the area of a Lv 4 hex
equal to 4.112 hectares.  This would grow enough 
food for ~1.5 extra people (in addition to the 
farmers).

  A Lv 5 hex = 28.78 hectares.  Or enough food to 
feed 10 extra people for a year. 


  TFT Wargame:
  Do I want to say that the LotU hexes are level 5
hexes?  55 meters across is a more reasonable size
for company sized fighting forces and it makes the
(very small) LotU map a bit bigger.  (1.21 km 
thru the mountain which is actually pretty 
reasonably sized.)  A level 5 hex which will feed
exactly 10 people (in addition to the peasantry) is 
nice.  
  However, it is FUN to say that one of the LotU 
hexes is a Lv 4 hex on my big map.  Helps keep 
things in perspective and I can (theoretically) 
lay out a company sized battle on my big battle mat.


  How does this sound?  When mapping Dwarven mines,
use Lv 5 hexes (or maybe Lv 6 hexes...).  This would
let us map pretty big areas on relatively few sheets
of paper.  Then for the interesting areas, do fine 
mapping of the individual rooms, etc. use the Lv 4 
hexes.  This level could show individual apartments.


  Looking at this from another direction, the scale
of the LotU hexes was picked so that individuals 
could be represented (and in the narrow corridors
they became important).  In a regular TFT war game
(with out 1 hex wide tunnels) individuals would
not really show up but an adventuring party of 4 to 
7 people would (especially if they had a fair bit 
of magic).

  Now companies in the Dark Ages tended to be a
size one man could control by shouted voice.  They
usually were smaller than Companies in the modern 
USA army.  If we say that:

Party  /  Squad   = 6 to 15 men.
?????  /  Platoon = 2 or 3 squads.
Maniple / Company = 2 or 3 Battles.
Cohort / Battalion = 2 or 3 Companies.

  This would make a company vary from ~30 to 110
men.  (Assuming if you have smallish squads you
put three squads into a Platoon.)

  Note that a Company will fit with room to spare 
in the Lv 4 hex, but things are too tight for a 
Battalion.  If we want the basic military unit for
the TFT wargame to be a Company of less than 100 
men (thus a PC party will be a noticeable unit) 
then this suggests Lv 4 hexes.


  Castles:
  Now castles varied widely in size.  Small forts
guarding a pass might be a dozen meters on a side.
Where as Himeji Castle ranged from 950 to 1600 m 
(east west) and 900 to 1700 meters (north south).  
(Himeji castle would take up more than a LotU map
(with the larger Lv 5 = 55 m hexes) by itself!  
However, that is exceptional.  Most castles would 
take a couple dozen Lv 5 hexes to map.  However 
large real castles become too big to map using the
LotU Lv 4 hexes (21 m).  Since a company sized TFT
battle system would want to be able to handle 
sieges, that is a reason to use Lv 5 hexes for it.


  *********************************************

  I am a bit torn.  I have been working on a TFT
war game and am trying to decide if the game 
should use 21 m hexes or 55 m hexes.  (This is 
assuming a LotU style, hex and counter game.  If
I went with a block game like Wizard Kings, I 
would want to use fewer larger hexes and then the
Lv 5 hexes become almost mandatory.)  What are 
people's thoughts?

  Maybe I should post some thoughts on my TFT
wargame for you all...

  ***********************************************


  Angle of hexes:
  If we assume that the Level one hex has a side
that faces north (that is the hex grain runs north
south and that the line which draws the north hex 
side actually runs east west), then what angle will 
the larger hexes have?  
  
  Well, from level 1 to Lv 2 is a 30 degree angle
(exactly).

  From Lv 1 to Lv 3 is a 10 degree angle (almost
exactly).  

  Ok, after some work on my big maps, going from
Lv 1 to Lv 4 is a bit Less than 30 degrees.  So
it will twist less as you go up sizes.

  As a rough approximation: Going up an odd number
of levels of hexes, the hexes will be tilted by
30 degrees.  If you are going up an even number of
hex levels, the hexes are not tilted.

  So if your 1 hex has the flat side of the hex
to the north (so that the line runs exactly east
west), then a Lv 5 hex (55 m) will also have a 
flat side that faces north.  
  If we go up to a Lv 8 hex (1000 meters) we could 
make its hex grain go east west (so a vertex  
will point north from the center of the hex.)
This will be fairly close.  


  ************************************************

  After doing all this work, I can see certain 
advantages to basing your mapping system on squares.

  Regards, Rick.
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