I have written many times about the "scalable rectangular rig" (SRR) This is a device for use within the conceptual massing enviroment which I find to be extremely useful. With it, you can apply 2 parameters to control the scale & proportion of an element, thereby achieving a great deal of variation very simply.

This can be compared to the age-old artists trick used to guide the eye and hand when scaling up (or down) an image. Alberti used a rectangular grid on a pane of glass to guide his transcription of nature on to a canvas marked out with a similar grid. I have no doubt that Michealangelo used the same method in reverse to scale up his sketches onto the surfaces of the Sistine Chapel ceiling.

I first used this method to when threading profiles onto a spline. The spline itself is hosted on a "ladder" within the rectangle. The width factor allows you to vary the curvature of the spline. Subtle variations in size and proportions are characteristic of the natural world. We used to call this organic form. Today it is fashionable to speak of biomimicry.

D'Arcy Thomson was a biologist/mathematician who looms large in any discussion of organic form. His book "On Growth & Form" influenced may architects & in it he used grids to illustrate the effects of proportional changes on biological form.

A variation on the SRR replaces the scale factor with 2 adaptive points. The result is a 2-pick component which will adapt itself to the points of attachment. Used with divide & repeat this opens up a wide range of possibilities. I typically use a scenario where we imagine ourselves as Oscar Niemeyer exploring variations on an idea for a cathedral at Brasilia.

The spline within an SRR can also be used to create a revolve. I first used this for an avocado pear, but more recently I devised a "form-finding" demonstration based on the gherkin to show how a wide variety of curves can be generated based on 3 or 4 variables.

All this is by way of background. Since I returned from RTC Auckland I have begun to experiment with circular rigs. These lack the "width factor" element, but have other interesting properties. Today I want to look at spirals.

Start with a "Generic Model Adaptive" template. Go to a plan view. Make a circle (reference line) Give it a radius parameter "R". Now draw a series of spokes from the centre point to the circumference, check "3d snapping" so that the ends are defined by nice black dots (reference points) Make sure you always draw from the centre outwards. Select all the points on the circumference and change the measurement type from NCP (normalised curve parameter) to Angle. Use these angle settings to space the points equally around the circle.

It's quite easy to thread a spiral around the spokes, just working "freehand" to guess the curve. I used spline through points which automatically operates as if 3d snapping and chain were checked. Change the radius to reassure yourself that the whole thing will scale up & down proportionately.

Now we just have to add precision & control ... plus some solid geometry that will show up in a project. It gets a bit tedious now (as in repetitive) Give each point a number (under name) this will reduce the potential for confusion. Then next to the NCP value, click on the little parameter button & give each point an instance parameter. Set point 1 to NCP_01, point 2 to NCP_02 etc. It does help to use the zero before the first 9 and to be systematic. It can be quite confusing if the parameter list is out of sequence. Try to get a rhythm going: name the point, add parameter, name it (you can use copy-paste and change the last digit), change to instance.

Next, create a number parameter called F (factor) you could also call it X if you like, doesn't matter. Set the value of F to one more than the number of points in your spiral. Now feed in formulae for the position of each point. F/(point no) All this does is to divide the length of each spoke into small equal lengths and step each point outwards by one of these divisions. I create more divisions than points so that the last point will fall short of the circle. Just in case I want to select it won't be confused with the point at the very end of the spoke.

You have created a geometric spiral. It's the kind of spiral you get when you coil a rope around on the deck of a ship. All the coils are spaced the same distance apart. You really need quite a lot of points to make a good geometric spiral because you want to go around the circle at least 4 times. I've been using 12 spokes. You could try using 8, but I think you will find that the spiral distorts, especially at the ends.

Most organic spirals are logarithmic. The distance between the coils expands as you go around. Think of a snail shell. You can think of it in terms of a graph. When the distances out from the centre increase by equal steps, that will be a straight line graph. Clearly the logarithmic spiral will give a curved graph. But we want to have the same start & end points (zero to one) Because that's the range we have available using NCP. Basic maths suggest we need to square something to get a logarithmic curve. Square of zero is zero, square of one is one. So we can square the whole thing & the formula for point 3 will look like this (F/3)^2

Make a second copy of your family and go methodically down the formulae, converting them to squared versions. Now you have two spiral families, one geometric & one logarithmic.

To create geometry we need two points hosted on the spiral, one close to each end. There are 2 reasons for keeping them away from the ends. Firstly so they are easier to select, secondly the spline distorts a little at its open end. The last part tends to straighten out just because there isn't another point pulling it back in towards the centre.

Place your points, make the workplane visible, place a profile on it. I have a few ready-made mass profiles sitting in a folder just waiting to be used. There are a bunch of them in my entry for last year's pumpkin competition. You can find them under downloads.

Link their radii to instance parameters. You need to make the inner one quite small. Select both profiles plus the spiral and create form. If it refuses, try a smaller radius.

I want to have the two profiles scale up with the radius of the whole spiral. So I use a little formula. I called my two profile sizes P1 & P2. Create number parameters X1 & X2

P1 = R / X1, P2 = R / X2

If you made everything with instance parameters, you can place a few of these families in a project and play with the variations. Scaling is easy. Just type in a radius and the whole thing scales proportionately. Vary the values of X1 & X2 to make the geometry thinner or fatter, more or less tapered.

One nice bonus is the ability to adjust the value of F. If you try to make is smaller the family will break, but if you make it bigger, the spiral will tighten up. Think of it this way. The outer end of the spiral is sitting at 25/26 of the spoke length. Almost at the end. If you increase F to say 50, it will move to 25/50. That's half way. The effect will be even more dramatic for the logarithmic spiral. Make small adjustments. The geometry will fail if it becomes self-intersecting. So if you want a really tight spiral you will have to creep up on it slowly.

It's quite easy to add an extra turn to your spiral. Just add 2 points on the next 2 spokes, select the spiral, control-select the points, hit spline through points. Repeat until you reach the length you want. Number the points. Add NCP parameters. Fill out the formulae.

I went on to play around with rectangular profiles a little. But by now I'm starting to wonder what the applications are. Sort of looks like wrought iron scroll-work but do you really want to get into heavy conceptual massing studies for a garden gate ? I guess you could imagine designing a water feature or similar landscape element this way.

Whatever. I had fun & kept my brain busy. If you want to play around with it you can download a couple of the families I made. Don't look too closely at the parameter names etc, they are just as they came out during my explorations.

spiral families

This can be compared to the age-old artists trick used to guide the eye and hand when scaling up (or down) an image. Alberti used a rectangular grid on a pane of glass to guide his transcription of nature on to a canvas marked out with a similar grid. I have no doubt that Michealangelo used the same method in reverse to scale up his sketches onto the surfaces of the Sistine Chapel ceiling.

I first used this method to when threading profiles onto a spline. The spline itself is hosted on a "ladder" within the rectangle. The width factor allows you to vary the curvature of the spline. Subtle variations in size and proportions are characteristic of the natural world. We used to call this organic form. Today it is fashionable to speak of biomimicry.

D'Arcy Thomson was a biologist/mathematician who looms large in any discussion of organic form. His book "On Growth & Form" influenced may architects & in it he used grids to illustrate the effects of proportional changes on biological form.

A variation on the SRR replaces the scale factor with 2 adaptive points. The result is a 2-pick component which will adapt itself to the points of attachment. Used with divide & repeat this opens up a wide range of possibilities. I typically use a scenario where we imagine ourselves as Oscar Niemeyer exploring variations on an idea for a cathedral at Brasilia.

The spline within an SRR can also be used to create a revolve. I first used this for an avocado pear, but more recently I devised a "form-finding" demonstration based on the gherkin to show how a wide variety of curves can be generated based on 3 or 4 variables.

All this is by way of background. Since I returned from RTC Auckland I have begun to experiment with circular rigs. These lack the "width factor" element, but have other interesting properties. Today I want to look at spirals.

Start with a "Generic Model Adaptive" template. Go to a plan view. Make a circle (reference line) Give it a radius parameter "R". Now draw a series of spokes from the centre point to the circumference, check "3d snapping" so that the ends are defined by nice black dots (reference points) Make sure you always draw from the centre outwards. Select all the points on the circumference and change the measurement type from NCP (normalised curve parameter) to Angle. Use these angle settings to space the points equally around the circle.

It's quite easy to thread a spiral around the spokes, just working "freehand" to guess the curve. I used spline through points which automatically operates as if 3d snapping and chain were checked. Change the radius to reassure yourself that the whole thing will scale up & down proportionately.

Now we just have to add precision & control ... plus some solid geometry that will show up in a project. It gets a bit tedious now (as in repetitive) Give each point a number (under name) this will reduce the potential for confusion. Then next to the NCP value, click on the little parameter button & give each point an instance parameter. Set point 1 to NCP_01, point 2 to NCP_02 etc. It does help to use the zero before the first 9 and to be systematic. It can be quite confusing if the parameter list is out of sequence. Try to get a rhythm going: name the point, add parameter, name it (you can use copy-paste and change the last digit), change to instance.

Next, create a number parameter called F (factor) you could also call it X if you like, doesn't matter. Set the value of F to one more than the number of points in your spiral. Now feed in formulae for the position of each point. F/(point no) All this does is to divide the length of each spoke into small equal lengths and step each point outwards by one of these divisions. I create more divisions than points so that the last point will fall short of the circle. Just in case I want to select it won't be confused with the point at the very end of the spoke.

You have created a geometric spiral. It's the kind of spiral you get when you coil a rope around on the deck of a ship. All the coils are spaced the same distance apart. You really need quite a lot of points to make a good geometric spiral because you want to go around the circle at least 4 times. I've been using 12 spokes. You could try using 8, but I think you will find that the spiral distorts, especially at the ends.

Most organic spirals are logarithmic. The distance between the coils expands as you go around. Think of a snail shell. You can think of it in terms of a graph. When the distances out from the centre increase by equal steps, that will be a straight line graph. Clearly the logarithmic spiral will give a curved graph. But we want to have the same start & end points (zero to one) Because that's the range we have available using NCP. Basic maths suggest we need to square something to get a logarithmic curve. Square of zero is zero, square of one is one. So we can square the whole thing & the formula for point 3 will look like this (F/3)^2

Make a second copy of your family and go methodically down the formulae, converting them to squared versions. Now you have two spiral families, one geometric & one logarithmic.

To create geometry we need two points hosted on the spiral, one close to each end. There are 2 reasons for keeping them away from the ends. Firstly so they are easier to select, secondly the spline distorts a little at its open end. The last part tends to straighten out just because there isn't another point pulling it back in towards the centre.

Place your points, make the workplane visible, place a profile on it. I have a few ready-made mass profiles sitting in a folder just waiting to be used. There are a bunch of them in my entry for last year's pumpkin competition. You can find them under downloads.

Link their radii to instance parameters. You need to make the inner one quite small. Select both profiles plus the spiral and create form. If it refuses, try a smaller radius.

I want to have the two profiles scale up with the radius of the whole spiral. So I use a little formula. I called my two profile sizes P1 & P2. Create number parameters X1 & X2

P1 = R / X1, P2 = R / X2

If you made everything with instance parameters, you can place a few of these families in a project and play with the variations. Scaling is easy. Just type in a radius and the whole thing scales proportionately. Vary the values of X1 & X2 to make the geometry thinner or fatter, more or less tapered.

One nice bonus is the ability to adjust the value of F. If you try to make is smaller the family will break, but if you make it bigger, the spiral will tighten up. Think of it this way. The outer end of the spiral is sitting at 25/26 of the spoke length. Almost at the end. If you increase F to say 50, it will move to 25/50. That's half way. The effect will be even more dramatic for the logarithmic spiral. Make small adjustments. The geometry will fail if it becomes self-intersecting. So if you want a really tight spiral you will have to creep up on it slowly.

It's quite easy to add an extra turn to your spiral. Just add 2 points on the next 2 spokes, select the spiral, control-select the points, hit spline through points. Repeat until you reach the length you want. Number the points. Add NCP parameters. Fill out the formulae.

I went on to play around with rectangular profiles a little. But by now I'm starting to wonder what the applications are. Sort of looks like wrought iron scroll-work but do you really want to get into heavy conceptual massing studies for a garden gate ? I guess you could imagine designing a water feature or similar landscape element this way.

Whatever. I had fun & kept my brain busy. If you want to play around with it you can download a couple of the families I made. Don't look too closely at the parameter names etc, they are just as they came out during my explorations.

spiral families

Applications? How about Volutes? I did a square rigging for mine, but now I want to try the circular one and see if I can get better results.

ReplyDeleteAnother great post. Thanks Andy.

Andrew, thank you for yet another informative, interesting, and nicely illustrated blog article. I love it!

ReplyDeleteOne comment: I wonder if you've had the chance to work with the parametric polygon. We made an exercise in the class about roofs, in Auckland, but on the second part of the class, when you were not there. I am going to send you the templates, along with a description and an exercise. I am sure you are going to find some nice applications for those templates.

Thanks Alfredo, yes I need to make time to go through the second part of your class which I missed at Auckland, because I found your methodical approach very helpful & as you know I am also very fond of shell roofs. Always a pleasure to swap ideas & share information.

ReplyDeletenice work.. I've studied spiral and helix too.

ReplyDeletehere is what i thought.

http://www.youtube.com/playlist?list=PLQMW6EuQeQZPI_T206yLFV-9dhyA5eier

Hi Andy,

ReplyDeleteI have just modeled a parametric tower mass in a "Generic Model Adaptive" file and "Load into Project".

The Mass Floors is grayed out and can't be accessed.

Do you know a trick how to make it work?

Thanks!