[Geezer (OP)]

OK - so I still play with toy trains. Get over it!

Here's the thing:

In the US, and most of Europe, a common model scale is HO, and the scale ratio from the real thing is 1:87. Due to a lot of history, the UK (mostly) landed on OO, which has a ratio of 1:76

Now, 1:76 against 1:87 does not seem like that much of a difference, but when you compare the same original modeled in both scales side by side, the difference seems to be far greater.

Is our perception of scale based on volume? If so, should we cube the difference in ratios?

[CliffordK]

You're in luck.

All you need to find is 1/76 scale models of the 4'1" or 4'0" gauge railways, and they should come out just about right.  []
http://en.wikipedia.org/wiki/List_of_rail_gauges#Medium_gauge

Metric	Std	Country	RR	1/87 width	1/76 width
1,435 mm	4 ft 8 ˝ in	About 60% of Worlds Railways	Standard Guage	16.49mm	18.88mm
1,245 mm	4 ft 1 in	England	Middleton Railway, pre 1881	14.31mm	16.38mm
1,219 mm	4 ft 0 in	Wales	Padarn Railway (1842–1961)	14.01mm	16.04mm
1,219 mm	4 ft 0 in	Wales	Saundersfoot Railway (1829–1939)	14.01mm	16.04mm
1,219 mm	4 ft 0 in	Scotland	Glasgow Subway, Falkirk (1905–1936)	14.01mm	16.04mm
1,219 mm	4 ft 0 in	England	Furzebrook Railway (c.1830–1957)	14.01mm	16.04mm
1,219 mm	4 ft 0 in	England	Redruth and Chasewater Railway (1826–1915)	14.01mm	16.04mm
1,219 mm	4 ft 0 in	New Zealand	Wellington tramway system: electric trams, system closed 1964.	14.01mm	16.04mm

[Geezer (OP)]

Ah yes, but (most) UK models are built to OO scale, but the track is really HO scale! It's not a big deal really, but it does look a bit unrealistic.

It's been that way for about sixty years, and it wasn't really a problem, because the UK railway system was isolated from Europe, so there were very few instances of mixed equipment operating on both sides of the channel. However, that situation is rapidly changing.

I have proposed to the UK community that it adopt a "sliding scale" based on the age of the prototypes so that OO would, over time, merge into HO. It's already half way there because of the track.

My proposal was not appreciated!

[Geezer (OP)]

Well, this one really lit up the switchboard!
 
It's really intended to be a (sort of) serious question about how we perceive things. Does using a one-dimensional ratio really give a true sense of what we perceive, or do we in fact perceive "scale" as more of a three dimensional ratio?
 

[CliffordK]

Yes & No

You may be right, for a space filling model, perhaps one should judge it by volume.

Consider a 50% model of a 10x10x10 cube (very exciting, I know), with a total volume of 1000 units.

If you go by length, then the resulting 50% model would result in a 5x5x5 cube, with a total volume of 125 units, or 1/8 of the above, or (˝×˝×˝).

If you tried to make it half the volume, then the length of side would be x L
Or, about  0.7937 x L, or nearly 80% of the original length.

So, setting three proportional cars next to each other.  The one that is 50% of the volume would be 80% of the length of the original.  The one that is 50% of the length of the original would be only about 12.5% of the original volume.

Which is best?

Apparently it gets much more complicated when making model aircraft, because lift is a function of Wing area (square), Cord Length (linear), and Velocity (Should velocity be reduced?).
Volume, and in many senses, mass, of course, is a cubic measurement.
Is drag proportional to the fontal area (a square)?

Anyway, to calculate the aerodynamics of the scale model, you have to incorporate linear measurements, square measurements, and cubic measurements. 

Large ships, of course, have a "hull speed" which is a linear measurement.  Yet displacement, and most other quantities are cubics.

[JP]

This reminds me of the experiments that show that people tend to misjudge the amount of liquid in glasses depending on the shape of the glass.  They tend to overestimate the amount in tall, skinny glasses as opposed to short, wide glasses. 

I guess what this means is that people (at least somewhat) extrapolate volumes from linear dimensions. 

[Geezer (OP)]

Yes - it's a bit weird.

For example, a 10% increase in scale (linear) does not sound like very much, but when you compare a two 3-D models, the difference seems much greater, which probably is not surprising because the volume has actually increased by 33%

As Clifford points out, dimensional scaling is only one aspect of the problem. Physical effects can't really be scaled.

[JP]

Yikes!  If the physics scaled linearly you'd be in trouble.  To run your model train, you'd have to use ~1/87th of the power needed to drive an actual train!  From a back of the envelope calculation (~1000 horsepower full sized train, 1/100th scale locomotive, and google conversion from hp to watts) you'd be running at ~7500 watts!

[CliffordK]

Yikes!  If the physics scaled linearly you'd be in trouble.  To run your model train, you'd have to use ~1/87th of the power needed to drive an actual train!  From a back of the envelope calculation (~1000 horsepower full sized train, 1/100th scale locomotive, and google conversion from hp to watts) you'd be running at ~7500 watts!

Ahhh...
All you need is Tim the Toolman Taylor...  to build you a super powered 7.5 KW micro bullet train 

[Geezer (OP)]

Yikes!  If the physics scaled linearly you'd be in trouble.  To run your model train, you'd have to use ~1/87th of the power needed to drive an actual train!  From a back of the envelope calculation (~1000 horsepower full sized train, 1/100th scale locomotive, and google conversion from hp to watts) you'd be running at ~7500 watts!

I think the rails would glow red hot and fuse! And 1000HP is a conservative estimate. The real things can produce several thousand kW. Scaling 3000kW by the cube of the 87 ratio gives about 4.5W which is actually pretty close to the power output of the electric motor in the model.

[Geezer (OP)]

I may have to complain to the NMRA that they are misrepresenting HO scale by more than a few orders of magnitude.

[SeanB]

I saw a train model for sale near me, but at 1.7m long the loco is a little big to fit in my room. The price is a big ouch too, around 1200 pounds.

[evan_au]

To properly scale the wind resistance, you would need to operate the model in a fluid other than air...