Bill Schopp, Trefoil Micro-Layouts, and Me in 2023*

Trefoil planform (according to Bill Schopp) shown in blue outline

There’s a brief note by Bill Schopp in the Trouble Shooting column of the May ’68 issue of Railroad Model Craftsman about a new idea for a layout table’s planform: the trefoil**. The idea was brought to Mr. Schopp’s attention by an enthusiastic reader identified only as ‘R.W.D.’. It seems Mr. Schopp was both intrigued and dismissive about the practically of the idea even though the reader said he was well into construction of a trefoil shaped layout. On the one hand Mr. Schopp notes his conflicting views:

“This idea held me enthralled for 3 days now trying to draw interesting layouts for it in N, HOn3, and HO traction with little success. I discarded at once the idea of ‘scenery first, track afterwards,’ as suitable for prototype but not model railroads: it’s the kind of idea an armchair expert could convince himself was the ‘only way’. I’ve drawn and redrawn the trefoil design with variations and have come to the conclusion that as a site for a layout, the trefoil table has little or no merit. But, I am constantly gnawed by the suspicion that I may be putting down one of the greatest ideas to hit model railroading since…well, since 2-rail.”

But then he firmly concludes that the trefoil planform probably is “not a good model railroad table” and goes on to list 3 issues he has with it:

1. “…it would restrict the design of a pike either to just following the curves, or making a 3-lobed version of a figure-8.”

2. “…takes up a terrible lot of space.”

3. Requires advanced carpentry skills to build.

In the Sept. ’68 issue a R. W. Bide of Lethbridge, Alberta wrote in to comment on Bill Schopp’s assessment,

“Thanks for not squashing the Trefoil. It’s the first sensible island design I’ve seen and I’m an around the walls layout fan. Two very good treatments for a start. Very high center of the lobe for scenery and a central valley full of action or very high ridges along the center lines giving three lovely scenes. Treatment could be a gorge or canyon, Lake Superior Shore or other Canadian Shield stuff. Or even a horseshoe curve. Walk around control with complete success. One lobe might be enlarged to the width of the others. The biggest problem I can see is radii but horrible curves can be hidden.”

When I read that letter I wondered if R. W. Bide was actually the mysterious ‘R. W. D.’ who alerted Bill Schopp to the trefoil planform. RMC didn’t have the most stringent copy editing in those days, so I wouldn’t be surprised if R. W. B. of Lethbridge was ‘R. W. D.’. I’ve looked in later issues for any mention of Bide’s trefoil layout, but I haven’t found any, so I don’t know how his worked out. If I find a story I’ll post an update.

Then in the Oct. ’68 issue a David Zavadi of State College, Pennsylvania wrote in to say that he derived the area of the trefoil plan and gave this equation***:

A = R² (6 √3 + π)

A is the area of the trefoil, and R is the radius of the circle upon which it is based.

Given the aforementioned copy editing issues I decided to see if I could still manage some high school level math and tried to derive that equation for myself. It was tough getting the brain going on these sorts of problems again after decades of neglect. I felt faint at times and had to lie down :-) but in the end I was able to confirm the equation is correct.

After getting this far I wondered what would R have to be for A = 4 sq.ft., which is the largest allowable area for a micro layout? The area equation needs to be re-arranged for R,

R = √ ( A / ( 6 √3 + π ) )

Plugging in A = 4 sq. ft. and evaluating the equation it turns out R = 6.5 inches. Ok, I should clarify the actual answer is 0.543699 ft, but I converted it to inches and rounded down a tad to get the more conservative and easily usable value of 6-1/2”.

4 sq. ft. trefoil compared to 2' x 2' foam square

I wanted to see how a 4 sq. ft. trefoil compared to a 2 ft. x 2 ft. foam square like the one I used to build the Loonar Module. So, I got out some paper, masking tape, scissors, compass, ruler, and protractor and went to work to make a paper trefoil. That's the result over on the left.

It looks like a very large fidget spinner.

4 sq. ft. trefoil overlaid on 2' x 2' foam square

Overlaying the two planforms gives a slightly better size comparison. 

Could this be the basis of a micro-layout, or do Bill Schopp's concerns squelch layouts in this category too?

I don't have a definitive answer. I think many of the points raised by Schopp and Bide still apply; however, I'll note that unlike back then we can build layouts from foam slabs, have wireless DCC control, have lots of specialized track and power trucks available, and it's not unusual for micro-layouts to be just simple track loops where scenes and scenery are the main points of interest. There need not be a concern with trying to simulate a fully operational, real railroad, or even a piece of one. So, maybe Mr. Schopp's concern with only being able to build a 3-lobed figure-8 track, or some such variation, might actually make for an interesting micro-layout. I look at the trefoil planform and think that it might be the way for me to build the Mt. Lowe layout I've wanted to do for years now. I can imagine each lobe providing a separate scene: circular bridge, granite gate, and the look out all visually separated by some suitable central rocky outcropping. 

4 sq. ft. 3-circle trefoil compared to 2'x2' foam square

If you look at this post's opening drawing with all the packed circles you can see that it's possible to slice out a three-lobed planform from the matrix based on just 3 circles instead of 4. This gives a less spindly and more compact planform.

The area for this shape is:

A = R² (4 √3 + π)

and the radius of the circle upon which it is based is:

R = √ ( A / (4 √3  +  π) )

If A = 4 sq. ft., it turns out R = 7.5" (again, I slightly rounded down R to yield a conservative and easy to use value).

4 sq. ft. 3-circle trefoil overlaid on 2' x 2' foam square

The overall shape is a bit blobbier than the 4-circle version, but it still has three distinct lobes. 

I've got to admit an advantage of the 2' x 2' x 1" pre-cut foam square is its convenience. I bought mine at Home Depot, but I should point out they are expensive for what they are: around $7 or $8 per square, which is crazy on a per sq. ft. basis compared to a 4'x8' sheet, however the pre-cut convenience can't be beat. If you're going to build either of these trefoils some cutting and gluing of foam slabs will be necessary. Or, maybe a base could be built up from corrugated plastic sheets. All this is just to say that some base construction will likely be needed, but likely no expert carpentry will be required as Mr. Schopp noted for his 1968 readers.

I haven't done a comprehensive scan of the internet, so I wouldn't be surprised if a lot has been done with trefoil planforms and their variants, so please excuse the naiveté on my part if this is obsolete news and it's just new and interesting to me.   

Digressions:

*What does this have to do with 1977? Well, nothing in the actual 1977, but in the alternate, phoney, time travelled to world of 1977 where I found this June 1977 issue of Rail Model Experimenter, a lot. Apparently that issue contained a lost article by Bill Schopp called, Advanced Trefoil Trackplans. In retrospect, I should have consulted that piece before even considering this post. If you have the issue, please let me know what's inside :-)

**The trefoil entry in Wikipedia shows shapes somewhat different than the ones in Bill Schopp's Trouble Shooting note. In this post I'm going to go with the nomenclature used in the old May '68 RMC story and leave investigating strict definitions for another time.

***The letter and the editor's associated comment mentions integration, polar coordinates, high symmetry, analytic expressions, "equasions", "plain" geometry, and so forth being deployed to find the area. For the trefoils shown in this post finding a closed form equation for the shapes' areas is not too hard, but what if you wanted to figure out if an irregular trefoil you were planning to use was less than 4 sq. ft. so it would still qualify as a micro-layout? It could be a tough go to calculate the area by developing equations for an arbitrary shape. It might be far more practical, and just as accurate, to use this method described in The Scientific American Book of Projects for the Amateur Scientist.