Does anyone know how to mathematically explain this SPIRAL?

I have been trying to play around with the spiral command for quite a while now, and the only way I know to make this spiral is to totally fudge it… I wish I could figure out how to do it mathematically somehow… the spiral command fails big time or I really don’t know how to use it.

my spiral failures:

my spiral fudging… and wayyy too much work, plus it doesn’t look clean:

Hi Izabella,
Fudging is my specialty. I start with a circle, split it in quarters, delete one quarter then use a combination of 3d scale and mirror on the working quarter. After mirroring each segment I scale3d by 1/3rd then mirror and repeat until the scroll is as tight as I require. I then join all segments, offset the curve the required thickness and cap each end of the paired splines. For the tapered inside point I hand draw a spline using the near snap to finish, trim the result, run the match command to get the tangent join. Finally join all segments then extrude as a solid. It all sounds long winded but is really quick once you have done it a few times. Also, Rhino for Mac has the handy right click command repeat so that speeds up the building sequence a fair bit. The beauty of this method is that once you have a spiral single line, you can save it and scale it to any size before running offset commands for larger sizes.
For a precise mathematical method, build a Fibonacci square with the 1,1,3,5,8,13 etc ratios and intersect snap your spline to the box corners.
I have done an article in our Blacksmithing magazine to demonstrate the method. Feel free to download a copy and have a read. ‘The Bellows’

Hope this helps. Cheers -Rob


i don’t really think it’s any of ‘the spirals’… maybe loosely going off a golden spiral:

the way it’s made probably has a lot of leeway with how perfect it needs to be… it’s being muscled into shape instead of numerical.

if you want exactly that piece, doing it by hand will probably be the way to go about it… tracing it sort of loosely then editing the points… maybe try to trace the edge instead of the centerline? it’s more of an exact target to aim for.

that’s what I suspected, but I was hoping someone knew something I did not:)
Thank you Jeff!

Although since these scrolls are ‘muscled’ into shape like Jeff correctly pointed out, your way works wonders Rob.

This is my first try:

and after some manipulation (scaling of each of the pieces) this is what came out:

No matter what else I scale more or less, it won’t fit exactly, but it’s pretty darn close.
Thank you so much!

PS. Don’t you wish someone would come up with some sort of way that you can free draw a scroll and it will automatically make itself line up, and then you can manipulate it a bit into place? That would be awesome!

oh wow! fascinating…
love the way you think! I’m about to try this… wish me luck:)
and thank you so much for sharing

so far, so good… looks pretty darn good… I’ll see if I can make it work on an actual design tomorrow:)
Thank you again!

Hi Dale…
Although the Golden spiral is Golden indeed… it doesn’t quite work in metalwork:(
Not most of the time anyway…
That was actually the very first thing I tried… I was super excited about it and failed miserably right after… haha.

Rob’s way works the best so far… it’s pretty darn good.
…so until someone else comes up with an easier way, I guess we’ll both stick to it:)

Rob, I get the feeling that you are an actual blacksmith who makes real parts. Have you been doing it since before CAD? I’ve seen spirals used a lot in metalwork over the years so I’m wondering whether blacksmiths just eyeball them into a pleasing shape or if they have a particular type of spiral and a method of obtaining it in mind (before computers)? Did they create templates to use as they are beating a bar into a spiral? If so, how did they develop them? Or does it involve a fixture to hold the end and the skill of the blacksmith in playing the applied bending force against the rate of cooling?

Just curious.

I do not actually do the forging, but I have been designing metalwork for about 10 years.
I’ve always done it on paper, then I started using Illustrator for certain things, and now I really would love to figure out how to translate this into rhino.
As far as the scrolls go, they all start out with the tip being forged down, that is placed into the starter of a scroll, and then they keep scrolling a spiral, but they manipulate the spiral to the shape of what is on the drawing… so they’ll expand it more or contract it more (muscle it into place.) You have to heat, manipulate, heat again, manipulate, and so on…
At the end of the day it takes an artistic soul to make them look good.

I have been an Artist Blacksmith for seven years. Blacksmiths use many methods to achieve scrolls, whether hand drawing with chalk on the floor and matching the iron work to the drawing, doing it by eye or they use a scroll jig. I have many jigs made up for loose scrolls and tight scrolls. I have been a Rhino user since it was a plug in for ACAD, so I have been using Rhino for all my ironwork designs. I can calculate material weight, length and center of gravity a lot easier than using a calculator and a piece of string or measuring tape. You also have the ability to generate a photo realistic image for a client that will sell a commission better than a hand drawn sketch.
Izabela is correct in that there is still tweaking involved in getting many parts into position on a complex design. The fiddling sometimes takes longer than the original forging. Even with the technology of today, many of the smiths in our association still like to have a little bit of variation in there scroll work so that it achieves that ‘hand forged’ look.

Rob, are you pretty comfortable with making like a formal French rail design on Rhino? (one of those lovely curved ones on a pitch… not straight balcony)
Is that possible?
I’ve gotten pretty comfortable with Chandeliers, but laying out stairs so far is a nightmare for me.

I have done a few spiral staircases over the years but I haven’t done one featuring ornamental ironwork. The standards in Australia preclude most designs for ironwork because of entrapment points. It is a fickle area to be involved in. I would typically model the actual architecture of the stair way, then model the wrought iron rail, framework and artwork to fit. The forging of the railing and framework is a little larger than I could cope with though. Generating stairs along a spiral spline is a little challenging and mathematical but can be done.

1 Like

We have one rule… no 4" ball may pass through any place on the rail, except by the treads.

Well… Since it can be done, I’ll keep trying and see what I can come up with.
If you have one you wouldn’t mind showing off, I’d love to see what can be done in Rhino:)