A New(?) Method for Drawing Spirals

[JHCC]

I realized the other day that one of the basic characteristics of a logarithmic spiral -- that for any point on the spiral, the angle between the tangent and the radius are identical to the comparable angle at any other point on the spiral -- could be used to draw such a spiral much more easily and precisely than the usual method of drawing quarter-circles of ever-increasing radius (such as on Gary Huston's inaccurately named "Laying out the Perfect scroll" video). I'm sure that someone between me and Archimedes has come up with this already, but since I've never seen it before, I made my own video showing how to do it. Here it is:

 

[Les L]

John, thank you for creating this video, this is the first time I have been able to understand how to lay out a scroll. The only ones I have made were “winged” and looked like it. 

[TWISTEDWILLOW]

x2 That’s pretty cool JHCC!
that makes it look so easy maybe even I could do it lol, 

Did that box of measuring tools you got the other day spark the idea or is this something you have been putting together for awhile? 

[Frosty]

Thanks John, that was very clear and easy to understand. I ran across this method online before but you did a much better job of demonstrating and explaining it. It's pretty straight forward if you leave out the mathematical proofs and explanations of why it works.

My only question was answered by watching you work. The length of the radius (is that the right term for a spiral?) increases from the starting point to the ending point of the line drawn. 

The only thing I'd do differently would be putting a pin at the center so I didn't have to think about it.

I've copied and saved your video in case something happens IFI and the bookmark doesn't work.

Frosty The Lucky.

[George N. M.]

John, very cool.  However, it seems to me that you have to pay attention to the increasing distance from the center point to spiral, e.g. each segment you lay out is X units further from the center.  Maybe you were doing that and I wasn't picking up on that step.

"By hammer and hand all arts do stand."

[JHCC]

The beauty of this method is that you don’t have to pay attention to the length of the radius/distance to the center point. As long as the blade of the protractor is touching the center point and as long as the vertex of the angle is at the end of the previous edge segment, you’re good. This is a no-math method. 

1 hour ago, TWISTEDWILLOW said:

Did that box of measuring tools you got the other day spark the idea or is this something you have been putting together for awhile? 

I’ve actually been thinking about this for a while. This idea started to come together when I saw this video from Uri Tuchman, and yes, one of the tools in that box give me the idea of using the protractor. 

1 hour ago, Frosty said:

The only thing I'd do differently would be putting a pin at the center so I didn't have to think about it.

I had the same idea, but that wouldn’t have gone over well when I was filming at the dining table. 

[Steven NY]

That is a great method! Thank you for the video. It really simplifies the process. 

Steven

[arkie]

John, VERY easy way to lay out a scroll.  Thanks for posting it!  Like Frosty, I bookmarked it immediately.  Drawing the squares for the Fibonnaci or Golden Mean is a pain in the backside.

Question, myself not being well versed on scroll/spiral curve generation, I realize that several of the different spiral curves look almost the same.  Is this the same or just similar to the Fibonnaci or Golden Mean curve?

[Frosty]

1 hour ago, JHCC said:

I had the same idea, but that wouldn’t have gone over well when I was filming at the dining table. 

How about a reusable adhesive like the picture hangers and a thumb tack point up? Yes, blunting the thumb tack wouldn't be cheating. I have a new glucometer so pricking myself to get a blood sample is no longer desirable. 

How about this for fun. When your spiral reaches the max width, reverse the angle and make an integral counter spiral. Perhaps lapping over and under at alternating intersections.

I'm going to have to dig out some roofing copper or a cookie tin. I haven't done chasing and repousse in a long time a spiral might be fun.

Frosty The Lucky.

[JHCC]

1 hour ago, arkie said:

I realize that several of the different spiral curves look almost the same.  Is this the same or just similar to the Fibonnaci or Golden Mean curve?

There is a strong family resemblance between all logarithmic curves spirals. The difference between them (in layman‘s terms) is how quickly the space between the lines gets bigger. For the golden spiral, the growth factor is φ, the golden ratio; that is to say, the length of the radius increases by a factor of φ (roughly 1.618033) for every quarter turn.

You can generate a golden spiral using this method by setting the protractor at 17° from perpendicular rather than the 5° shown in the video. I mentioned this in the video, but I referred to it as the Fibonacci spiral rather than the golden spiral.

Strictly speaking, a Fibonacci spiral is not 100% identical with the golden spiral, but it’s a pretty close approximation. You create a Fibonacci spiral by combining a sequence of quarter circles with radii equaling the numbers in the Fibonacci sequence: 1, 2, 3, 5, 8, 13, etc. Because the ratio of adjacent Fibonacci numbers approaches φ the closer you get to infinity, the closer a Fibonacci spiral will be to a golden spiral the farther out you draw it.

[Irondragon Forge ClayWorks]

You mean we are not supposed to layout our scrolls freehand, like I have been doing all those years.

[JHCC]

Whatever works, man. Just one more tool in the toolbox.

2 hours ago, Frosty said:

How about this for fun. When your spiral reaches the max width, reverse the angle and make an integral counter spiral. Perhaps lapping over and under at alternating intersections.

I’d suggest marking equal intervals around the circumference of a circle and spiraling in (clockwise and counterclockwise) from there. 

[arkie]

John, thanks for explaining the differences in the curves and noting the angle for the Fibonacci curve.

[JHCC]

By comparison, an arithmetic (or Archimedean) spiral has a constant distance between the lines, which makes it look a lot more circular. 

Very generally, Archimedean spirals have been known since antiquity, while logarithmic scrolls have only been mathematically defined since the Renaissance (first described in 1525 by Albrecht Dürer and later studied by Descartes and Bernoulli). This may be while you see a lot more Archimedean spirals in metalwork up to the 17th century:

While the fashion changes to logarithmic spirals in the 18th century:

I don't have the images in front of me, so take this with a grain of salt, but if memory serves, Samuel Yellin would sometimes take a medieval piece for inspiration, but tweak the scrolls from Archimedean to logarithmic to make them more pleasing to the eye.

[TWISTEDWILLOW]

Wow that last one looks like it took awhile! I wonder how many hours and how much fuel went into that? 

[JHCC]

At Quad-State 2019, I saw Aislinn Lewis and Mark Sperry of Colonial Williamsburg make this sign bracket in about 12-14 hours of forge time. However, that included a LOT of lecturing and Q&A, so maybe 2/3 of that (or less) was actually working in the piece.

In a large shop with a bunch of journeymen and apprentices working under a master (or even several masters), I imagine that gate could have gone surprisingly quickly. 

[JHCC]

A further thought: there's nothing that says one has to use a locking protractor. One could just as easily use a piece of thin sheet metal with the proper angle cut into it or even a couple of strips of cardboard stapled together.

[Frosty]

A few years ago I downloaded a program for my comp to produce scrolls. It was hard to use and of limited use so I stopped messing with it. 

Frosty The Lucky.

[Rojo Pedro]

Very cool. Some of the hardest survey calculations include spiraled curves or railroad curve or highway curve as they are sometimes called. 

Our state DOT doesnt use them any more as it adds a lot of difficulty to construction and a bit of ambiguity to rights-of-way lines (ownership)

I will have to look but I thought a railroad curve had an increasing radius and a floating radius point

[JHCC]

I’d never given any thought to the specific geometry of railroad curves, but some quick googling turned up the Wikipedia article on track transition curves, which I will have to read more carefully later on (perhaps once the caffeine levels in my system reach their minimum acceptable level).

 In the mean time, you’ve gotten me thinking about the possibilities of using this method with some kind of variable reference rather than the fixed center point. Something for further exploration. 

[anvil]

I had a couple years in engineering and had a fair amount of calculus, but boy howdy i certainly missed that! Thats very cool,not to mention mind blowing.

 

 

 

 

[arkie]

anvil, I spent my first years in college as a math major.  I had reams of calculus courses, diff, integral, etc. etc...I subsequently changed majors to geology.  I NEVER, NEVER had an opportunity or need for the many hours of calculus I took.  What a waste of time!  Computers did all the math.

[Irondragon Forge ClayWorks]

Calculus and high school in general kept me out of college. However not all knowledge comes from college.

[JHCC]

I took calculus my freshman year in college to help fulfill my distribution requirements. The teacher was so dynamic and engaging that I ended up taking two full years of calculus with him and even took a semester of discrete mathematics and graph theory. I’ve never used the math since then, but some of the concepts have been tremendously useful. It’s also what introduced me to the Fibonacci sequence, which was definitely not a waste of time. 

[ThomasPowers]

What's fun in computer science is that you have to take the calculus courses and then you have to take the courses on Numerical Methods  to learn how to fake a lot of the absolute answer methods using lots and lots of smaller calculations.  I had some math majors in my NM class and it would drive them crazy when the prof would say that an approximation was as good as the "right" answer---if the approximation is close enough!  (2x2=3.999999999999999999999999999999999 is pretty much close enough for anything I do!) We also went over how to tell how close we were to the "right answer".

Another class that can be fun or can be horrible was Probability and Statistics; the text and teachers I had went over how the early work in P&S was all done by the old time math greats with respect to *gambling* and estimating their chances of winning...My text was full of real world examples like estimating from partial information.  During WWII the allies wanted to know just how many tanks Germany might have and so since every tank had a sequential serial number, every time they killed a tank they would take the number and send it back to the math folks who could create a bell curve on how many tanks it was likely were made.  After the war they got the tank factory records and found their best guess was something like within 10 tanks of the actual number.

Such classes can be dull memorization of formulas or fun figuring out the possibilities of craps!