From: throopw@sheol.UUCP (Wayne Throop)
Newsgroups: talk.origins
Subject: Holden Numbering
Summary: explanation of the Holden Number, and why Ted gets it wrong
Date: 17 Aug 92 02:16:17 GMT

> From: rh@smds.com (Richard Harter)
> Message-ID: <1992Aug16.023113.3121@smds.com>
> As far as "Holden's number" is concerned, it reminds me of the old
> chestnut of the three men, the hotel, and the bellboy.
> [...]
> The trick, of course, is that numbers that don't go together are being
> combined illegitimately.  It's all a bit of flim-flam.  Holden's number
> is much like that missing dollar...

Well...  the metaphor is appealing, but isn't sufficiently exact for my
taste.  The point I'll try to bring out below is that Ted (though he
explains himself very poorly) is not adding apples to oranges as in the
hotel bill, but rather he is keeping a separate basket of apples that he
refuses to add into the total. 

First, just what is the Holden Number?  It is the weight in pounds-force
over the square of the cube root of the mass in pounds-mass mass of
any given creature.  So

                     lbs        HN
                +---------------------
        cat     |     10        2.15
       human    |    200        5.85
      elephant  |  10000       21.5
    brachiosaur | 150000       53.1

What does this calculation mean?  It is a measure of force (the lbs
weight) per available cross section (the lbs mass ^(2/3)).  The
lbs mass cube-rooted and then squared because the mass is a rough
indicator of volume, and hence the ^2/3 power of mass is a rough
indicator of cross section.

Whence then the Holden Limit of 27?  Well, if an animal is lifing
more than its own weight, we can simply add the additional force
to the top of the ratio, yielding a Holden Number which characterizes
that lift.  Holden claims that Kazmaier's 1000 lb deadlift is the
largest lift that any creature can perform, and thus he calculates

         (1000 + 350) / 350^(2/3)  ~=  27

to get a figure for the maximum possible Holden Number of any creature.
Since this is just beyond the actual Holden Number for elephants,
Ted then jumped to the conclusion that elephants are the largest
creatures that can exist.

There are many side issues, such as

     - smaller weightlifters than Kazmaier have higher Holden Numbers
       for their weightlifting acts
     - other primates can sustain weights far larger
     - Guiness (despite some doubt as to the accuracy) lists an
       elephant beyond Ted's limit
     - elephants in circuses have been reported doing acts with
       HN larger than 30

Ted has explanations for all of these points, but showing why Ted's
explanation of each is incorrect would take a large posting on its
own for each of these points.  Perhaps another day.  These issues
are small potatoes anyhows.

What I want to pursue instead is what I consider the three primary
reasons that the Holden Limit is incorrect.  (This is a slightly
different list of three than I have pointed out before: I've slightly
rearranged the "tiling" of the three points, but they still cover much
the same area.)

     1)  biped/quadruped differences
     2)  proportional scaling differences
     3)  leverage differences

We can correct the Holden Limit to apply to quadrupeds by simply adding
the weight Kazmaier can lift with his arms to that he can lift with his
legs to get a rough idea.  That modifies the formula to

                (1660 + 350) / 350^(2/3)  ~=  40

Still short of the HN for a brachiosaur.  But we can account for
proportional scaling differences between species by dividing by a
more appropriate factor.  We can't really determine what that
factor is easily, but we can get a ballpark idea by using the
bone thickness factor reported by Alexander and others of (1/2.73).
This might give us a different scale completely, so we have to do the
figure for both Kazmaier and the brachiosaur:

             (1660 + 350)  / 350^(2/2.73)    ~=  27.5
               150000   / 150000^(2/2.73)    ~=  24.2

( By coincidence, the limit number here turns out to be much the
  same as the original Holden Limit. )

So, if we simply take nonproportional scaling into account, as Ted
refuses to do, we can see that at least the brachiosaur is within
the corrected limit.

We can add a factor correcting for leverage to our formula by assuming
that limb length scales as cube root of mass, while joint radius scales
as 2.73 root of mass.  Working it out, we get this (note that the
limit this time is 22):

       (1660 + 350) / (350^(2/2.73) * (350^(1/2.73)/350^(1/3))) ~= 22.6
    600000 / (600000^(2/2.73) * (600000^(1/2.73)/600000^(1/3))) ~= 22.6

That is, the corrected limiting size of an animal in 1G is about 600000
lbs.  The largest plausible figures for sauropod masses are half that. 
At large sauropod sizes, this means that limbs only about 20 percent
proportionally thicker than Kazmaier's are required.  And such legs
would, indeed fit under a sauropod.  At this point Ted objects that
thicker legs would cause such a creature to become "muscle bound", but
this seems very unlikely to be a significant factor for such a small
thickening as 20 percent. 

Now, Ted's resolution of this "problem" is different.  Instead of taking
into consideration a more detailed model, Ted persists with the
oversimplified Holden Number, exploits his confusion between the
slimmest of sauropods (the diplodocids) and the stoutest (the
brachiosaurids), combines this with the mistaken impression that
diplodocid masses are calculated by force estimation and the
brachiosaurid masses by volume estimation, and comes up with the curious
formula (against the original Holden Number scale remember, where the
limit is 27, not 22):

          150000 / 500000^(2/3)   ~= 23.8

Both methods account for sauropods.  Ted does it by assuming
that sauropods were subject to 150000/500000 G effective acceleration,
or about one-third gravity.  Why Ted's attempts to supply a reduced
effective acceleration fail, of course, is material for several
additional posts.

As for me, I can't see that the .3 gravity is required to solve the
"problem", because the assumptions behind the Holden Limit at 10 tons or
so in 1G are simply incorrect. 

Anyway, I think you can now see why I'd say that Ted is adding apples
to apples... his calculations have the correct dimentionality (though
Ted himself calls the Holden Number "dimensionless", which is, of course,
another error on his part).  He just isn't adding up all the apples
he has, choosing to ignore those in his "quadruped", "limb proportions"
and "leverage" baskets.
--
Wayne Throop  ...!mcnc!dg-rtp!sheol!throopw
Newsgroups: talk.origins,soc.culture.african.american,alt.rap