## A false sense of symmetry

Although our initial plan to light the petals was to put waveguides between the petals and LEDs placed on a flat rigid PCB (see “About LEDs“), Alexis recently told us using flexible PCBs might be possible.

Flexible PCBs could be placed directly on the inside of the demi-sphere of the sculpture, thus avoiding the use of waveguides and ensuring a good luminosity. Their drawbacks however includes a high cost and the fact that Alexis hasn’t yet used them for previous projects.

Using flexible PCBs, our idea would be to take advantage of the symmetry of the sculpture and place identical PCBs along each of the 13 spirals. On each PCB, we put LEDs corresponding each to a petal. There is one catch though: the spirals are all slightly different… Indeed, it is an important property of John Edmark’s sculpture to have each petal placed at a unique height and angle on the sphere, with a unique size. While this is great for fluid animations (see Generating 3D Models), it makes the placement of the LEDs on the PCB a bit of a headache…

## LEDs placement

We decided to use a smaller version of our sculpture, with only 8 clockwise spirals and 13 anti-clockwise spirals (The previous images we showed you were of a 13 and 21 spiral model)

Ideally, we want each LED to be placed at the center of each petal. But using an identical LED placement on 13 different spirals will obviously introduce off-centered LEDs: our goal is to minimize this. In order to do this, I first aligned all of the spirals in the same orientation, and sorted them according to their lowest petal height:

The petals are getting in our way here, what we want is to just see the quadrilateral surrounding each petal.

We then need to flatten everything, it will makes things even clearer, and the PCB schematics needs to be done in 2D anyway.

The floor you see on this image is the actual floor of the pedestal on which the sculpture lies. You can see some petals are partly buried under the floor: they still need to be lighted for a fluid animation, but some are unfortunately too small to have LEDs placed underneath them. With this additional constraint in mind, I spent some time in blender figuring out the best LED placement, and this is the result:

Here is a superposition of the worst case spirals, which was also very helpful to design the placement:

Alexis is currently gathering information to assess the feasibility of flexible PCBs for our project, we will make sure to keep you updated on our final choice.

## Choosing our components: “We are dwarfs on the shoulders of giants”

For our project, we need to test our components in order to find the best way to control our marble. Good news, our project has been done in other ways before. That’s why we take a look at this webpage: https://wiki.fuz.re/doku.php?id=projets:datapaulette:1bit_textile (in French)

### Diameter of the marbles

Our first idea was to choose the smaller marbles, which have a diameter of 5mm. I have some of these marbles at home, so I tried to make a prototype with some cardboard.

Conclusion: my biggest fear was that a marble would move the other marbles when we rolled it (because of a too big magnetic field), but with my prototype, they don’t. If no, we would have to always create a magnetic field with our coils and that’s impossible. We would have to change our project for toroids, which are uglier. The other thing is the fact that 5mm marbles are not really easy to roll. That’s why we choose to order 6mm, 8mm and 10mm marbles for our test

### The coils

The other project also uses coils. The main difference is the fact that our coils are not hand-made, and our coils are under the marbles, not around them. However, they should have the good range for the coil. We just have to do a little calculation we learned when we were in Classe Préparatoire, with the value given by the other project.

60 turns, 50mA, 6mm circle and a distance equal to almost 0: the coil is about 0,6mH

With a 0,1mm diameter wire and a typical resistance of 5A/mm², we find an RMS current of almost 400mA

We choose bigger coils because they would ask less current from our processor for working. We finally ordered ten coils of 3mH, ten of 1mh and three of 680µH (the last stock, if we choose theses ones, we would take an equivalent) for our test

### Hall effect sensor

For this sensor, we have to choose which type we want. Understanding the different type is not easy, but you can find plenty of web articles about this. The best thing is to have a linear sensor because we will have a better idea of the state of a marble. When it will be on one way, the field will be negative and the sensor’s tension under 1,75V and on the other way, it will be positive and the tension over 1,75V. We also choose a directional sensor that gives the field in the three directions of space. It might be harder, but we could choose only one sensor for four or nine marbles if it works well and if the fields are easy to distinguish

### I/O expander

Finally, my last contribution for today is the I/O expander. First, we look at the possibility given by the project Datapaulette: shift register. After a short time of reflexion and the help of Alexis, we understand that it was a bad idea because we cannot control as much as we want the current in the coils. We will have to choose an FPGA that will give us enough I/O for our device. We need a total of 2048 I/O pins (one for the value of the Hall effect sensor and one for the current on the coils, the others will be linked to the ground or to VCC). We will wait for the end of the tests to choose our expander, but it should be one of these expanders. We just have to choose which number of ports we want by expander.

## First ideas

Our original idea was to place LEDs on the inner sphere of the sculpture, either with flex PCB, or by drilling the sphere, placing the LED in the holes and connecting them with wires to a rotating PCB contained into the sphere. To facilitate the positioning of the LEDs, we could have modified the design so that we can pin the petals one by one on the inner sphere rather than print everything in one block.

But these designs are not easily achievable. First, Alexis does not know how to design flex PCBs. Second, to have a satisfactory visual impression, we would like to have at least 100 petals. To justify this number, here is a model with 60 petals, and another with 100 petals, and a third with 150 petals.

With 60, 100 and 150 petals respectively.

In the above images, there will also be petals all the way to the center, but there will be only one light for all of them. In the above images, only the petals that would have their individual LED are shown. There will of course be petals all the way to the center, but the center ones will all share the same light.

With 100 petals and a single LED per petal, it would make 400 solder wire, which is unthinkable (especially given that the sculpture will turn).

## Flat PCB

After discussion with Alexis, we agreed to place the LEDs on a flat rotating PCB with the same phyllotactic arrangement. Here is a simulation example to imagine how we will place the LEDs:

The idea of abandoning 3D sculptures without struggle did not please us. That’s why we will try to give a 3D impression while leaving the LEDs on the flat PCB. To do this, our best idea so far is to use light pipes to guide LEDs light to the petals, like in the figure below:

Ideally, we would be able to keep flower-shaped sculptures similar to those of John Edmark. But in case we cannot have full-blown 3D sculptures, we could still add a little bit of  relief. Here’s what it could look like:

Today, Alexis has ordered a selection of compatible light pipes and LEDs, so that next week, we can do tests to measure the loss of brightness per cm inside the pipes and the influence of a LED on its neighbors (loss of brightness at the entrance of the guide).

## LEDs overheating

Another problem that we are likely to encounter is the overheating of the LEDs. For 100 RGB LEDs (which need to be switched on at the same time), with a maximum of 100 mA per color, a current of 30A is required. So if we power the LEDs with 4V, we need a power of 120W.
We started to think about solutions to cool the LEDs: drill holes in the sculpture to ventilate the LEDs, use a Metal Core PCB (MCPCB, such as below) instead of a FR4-PCB, use heat sinks, LEDs with an electrically isolated thermal path (such as Cree XLamp LEDs).

Layers of a MCPCB

But our LEDs will be under light pipes so they will not be well ventilated by the holes in the sculpture. The MCPCB will not allow us to put all the components that we need, in particular we can not solder components on the back of the PCB. The mask solder is only on the top layer, on which the LEDs are placed, and the bottom layer ends with aluminum or copper, and it is not possible to add FR4 and copper after the layer in aluminium. And the heat sinks all seem to have a disproportionate size for our use.

Before continuing research on the overheating of LEDs, we will use LEDs already ordered which I mentioned above to test their heating.

If you have any idea or suggestion, all comments are much welcome !

## TL;DR

• We need to generate 3D models of phyllotactic patterns.
• We give an explanation of how to generate phyllotactic patterns on a sphere.
• We present an issue we encountered, and the solution we found.

## Why 3D Models ?

When we first started thinking about the project, we quickly realized we would need to generate 3D models of the sculpture ourselves.

First because we need to have full control on the model, to try various configurations for the future 3D printed sculpture. And second because it will greatly help us visualize all the kinds of animations we are imagining.

Accordingly, Vlaya wrote a script using the blender python API to generate a 3D model, following John Edmark’s online explanation.

## Phyllotactic Patterns on a Sphere

We place the first point on the “equator”. Then to place each following point, we rotate 137.5° and move a little closer to the top of the sphere.

The big question is : how much closer is “a little closer to the top of the sphere” ?

137.5° is the golden angle, derived from the golden ratio. It has the nice mathematical property of being as hard as possible to approximate with rational numbers. This allows it to waste as little space as possible, which is why plants use it. And it looks nicer. You can try the simulation on this page to get a better grasp of this.

## A First Attempt

The first thing we tried was to actually place the points on a cylinder, rotating them 137.5° and going up by a fixed amount each time, and then projecting those points onto the sphere :

But we soon noticed an annoying discrepancy between what we got, and what John Edmark’s Blooms look like :

Our petals get more and more squashed as we near the center.

A consequence of that is, if we follow a spiral from the center outwards, at some point we’ll realise that what we’re following isn’t a spiral anymore.

All this doesn’t happen with John Edmark’s Bloom.

The same thing actually happens in sunflowers:

But in our case, it happens because the points step too slowly towards the center, whereas in a sunflower, they step too fast.

This paper on phyllotactic spirals gives a good explanation.

## A Better Solution

Let’s formalise things a bit:

A point on the sphere can be located by two arcs of a circle : parallel arcs (“horizontal”) and meridian arcs (“vertical”). To rephrase the previous explanation, we place each following point by moving along a parallel for 137.5°, and then moving up along a meridian for some arc length.

As we get closer to the top, the parallel arcs of circle become smaller.

The meridian arcs also need to get smaller, otherwise we would overshoot the top.

A bit of geometry yields this differential equation, where a(x) is the “vertical” angle between point x and the horizontal plane, and k is a coefficient:

``````a'(x) = k cos(a(x))
a(0) = 0``````

The solution is:

``a(x) = Arctan(sinh(kx))``

We had to tweak the value of k a little bit to get nice results.

Here’s what it looks like:

## Flat Circular Phyllotactic Patterns

We can do the same thing with flat circular patterns : place each following point by rotating 137° and moving closer to the center. The question is still “how much closer ?” and the geometry is merely a bit different.

We get this differential equation, where r(x) is the distance of point x from the center, k is a coefficient and R is the radius of the circle:

``````r'(x) = - k r(x)
r(0) = R``````

The solution is:

``r(x) = R exp(-kx)``

This gives us flat patterns like that:

Stay tuned for upcoming 3D animations !

## Ah, yes. Stereoscopic vision.

This is a follow-up to my previous article, in which we explored the choice of PCB placement in order to get an optimal visibility.

My previous analysis only covered the case of a punctual point of view. However, most people have two eyes, which provide a double point of view and could increase our coverage of space. It is to be pointed out that a point of space seen by one eye does not provide depth information to the brain, however let us assume that our image perception is smart enough to overcome this limit.

In the following simulation, we used a typical eye distance of 6 cm. To understand the different configurations, please refer to this article.

As you can see, the dispositions that seem to benefit the most of stereoscopic vision are modulo forests, particularly the ones where the repartition of PCBs seems the most uniform. The parallax effect is all the stongest when PCBs hiding each other are far away in the axis of vision. Modulo forests with k = 3 and 11 seem to be good candidates as the front portion of the cylinder is (almost) complete in both cases.

## Choosing a shape

As stated in this previous post, we have to compare different PCB dispositions to avoid blind spots. We soon came to realization that we would not be able to get rid of blind spots with our design, but it is possible to study the different ideas we came up with in order to mitigate this issue.

That’s why we created a python program that can simulate :

• blind spots created by PCBs hiding each other
• the variations of brightness due to the fact that LEDs can be seen at an angle (which is what created the dark center zone in CyL3D, which we are trying to get rid of)

## Stairs : an iteration over CyL3D’s design

To avoid the issue of a dark zone due to all LEDs facing the observer at an angle in the center zone, an idea was to make sure that all LEDs were not facing the same direction. There came the stairs idea:

After modelling this idea in python we get:

As you can see, there are a lot of deadzones, particularly in the central viewing axis.

Imagine a configuration with one vertical PCB with LEDs on both sides, facing the radial direction. Wait, don’t. I modeled it so you don’t have to :

This creates a 2D cylindrical POV display, with dark zones on the sides, which is in our opinion much better than a center dark zone. To create a 3D POV display, we “only” need to add concentric cylinders. The question is now: how serious is the problem of PCBs hiding each other and how to avoid it.

## Spiraling down the rabbit hole

The idea of spirals seems natural. PCBs can laid out with an angular offset when compared to the adjacent PCB. For balancing reason, we alternate the PCBs with a central symmetry in order to create a double spiral.

The major issue is that the adjacent PCBs tend to hide each other in large portions of space, due to the fact that they are aligned and closed to each other. We can also notice that the resulting shape is far from being symmetric (which is expected with a spiral).

## Some arithmetics

In order to shake things up a little more, we can keep the idea of a constant angular offset, but with a twist.

Let us take a circle split in n angular positions. If you choose a starting point and then jump from position to positions by skipping k positions, you will reach all positions if, and only if, n and k are coprime. This gives us a more evenly spread disposition.

For the sake of convenience, and because it is a nice name, we shall name this configuration “modulo forest”.

Deadzones tend to be much more spread out, which seems more acceptable. Further investigation has to be done in order to choose a definitive shape.

## Design Crisis

We discovered our first model contained a lot of issues. The outermost PCB often hide PCB behind them as you can see below. This results in some voxels being invisible. So we decided to create a simulator on Python to find the invisible areas. We will use it to determine the optimal configuration for our display. This configuration has several parameters such as the number of PCB, their position and the arrangement of the LED on one or both sides.

This a an example with 8 double-sided PCB in double spiral and 100 angular resolution. The blue dots are the visible voxels. The red dot is the user point of view and the green dot is the rotation center.