Glowing Polyhedrons

Building wireframe polyhedra¹ made from LED filaments, using graph theory to devise geometry and driving strategies.

It all began with a video by Huy Vector² that someone posted on cnlohr’s Discord server: a brass wire cube with a battery and four white LED filaments. I was immediately fascinated by the idea of building objects out of LED filaments.

But of course, in my mind, this quickly turned into a logic puzzle: why only four filaments when a cube actually has twelve edges? What if we built the entire cube out of LED filaments? How could we ensure that all edges light up properly? What about more interesting shapes?

Turns out this is a deep rabbit hole. In this article, I summarize some of my findings, the tools I built, and which filament objects I found to be suitable.

LED Filaments

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The LED filaments commonly found in Asian online marketplaces typically measure 38 mm in length with a forward voltage of 3 V. Each filament consists of 16 blue LED dies mounted on a thin white ceramic substrate. All LED dies are connected in parallel. Metal contacts at both ends serve as anode and cathode. The entire assembly is encapsulated in a silicone coating laced with a phosphor, which converts the blue light into a broader spectrum of colors. Blue filaments use a clear encapsulation without phosphor, allowing us to see the individual LED dies inside.

The filaments are diodes: current can only flow in one direction, from anode to cathode. When a forward voltage of about 3 V is applied, the filament lights up, with brightness proportional to the current flowing through it (typically 10–100 mA). When multiple filaments are connected in series, their forward voltages add up. Connecting them in parallel divides the current among them. I stored a more detailed characterization here.

Building wireframe shapes

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With some care (and a lot of patience), the metal ends of the filaments can be soldered together to form complex 2D and 3D shapes. This can be used to build amazing glowing objects. The challenge is that the electrical circuit is also defined by how the filaments are connected in the object; a mesh that results in a mechanically stable and visually interesting structure may not necessarily represent a circuit that allows all filaments to light up properly.

How do we solve this? Let’s start with a simple 2D object. The photo below shows a simple square made from four LED filaments. A constant-current supply is attached to the top and bottom joint, allowing all filaments to light up. The left and right (blue and red) parts of the circuit form two parallel paths for the current to flow from top to bottom, while the filaments on each side are in series. The total voltage drop is around 6 V, and each filament receives about half the total current.

As shown in the figure on the right, the structure can be represented as a graph: each junction corresponds to a vertex, and each filament corresponds to a connection between two vertices, a directed edge. Yellow markers indicate the feeding points where current is injected. This abstraction allows us to analyze the relationship between the geometry and its electrical properties using graph theory.

What do we want to achieve?

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Given a wireframe object made from LED filaments, what do we actually want to achieve? Some straightforward objectives:

1. All edges shall light up.
2. We want to minimize the number of feeding points \(P\), the vertices where we connect the power supply.
3. The path length for all circuits between feeding points shall be exactly \(L\) edges.

The third condition is needed to ensure driving with a constant voltage supply. The voltage drop between feeding points will then be \(V_{tot} = L \cdot V_f \approx 3V \cdot L\).

The square above meets the first condition with two feeding points and the path length is \(L=2\). Things get quickly more complicated when we move to more complex objects.

The Cube

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Let’s explore a simple 3D object first: the cube. A cube has 8 connection points (vertices, \(V=8\)) and 12 filaments (edges, \(E=12\)). Each vertex connects three edges, making it a 3-regular graph. There are trivial solutions for \(L=1\) and \(P=8\), where each vertex is connected to a power supply with alternating polarities. The images below show a solution as a 3D graph and a flattened 2D representation (Schlegel diagram). The number next to each edge indicates the normalized magnitude of current flow.

Unfortunately, there is no solution that allows fewer feeding points while maintaining a constant current through all edges. However, there is a solution for \(L=3\) and \(P=2\), where the feeding points are connected to opposite vertices of the cube. The images below show this solution as a 3D graph and a flattened 2D representation. We can see that, while all edges are lit, the current distribution is uneven: Edges in the middle of the path only carry half of the current due to branching.

Below you can see a photo of the actual cube with \(L=3\) and two feeding points (\(P=2\)). While the current varies by a factor of two between edges, the impact on the appearance is surprisingly small and is rather exaggerated in the photograph. The reason is that we tend to judge the filaments in isolation rather than against a constant brightness background. The eye’s response to luminance is less than linear, and it is mostly sensitive to relative brightness differences of objects next to each other, so the filament-to-filament variation is less noticeable relative to a constant background.

The Octahedron

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The octahedron is another simple polyhedron with \(V=6\), \(E=12\). In contrast to the cube, each vertex connects four edges, making it a 4-regular graph.

DC Driving with 2 Feeding Points

No solution exists for a simple DC driving scheme where bias is applied to only two vertices (anode and cathode). As shown below in the first two images, only 8 out of 12 edges can be made to light up when driven this way.

Multiplexed DC Driving with 4 feeding points

However, as shown in the rightmost image, if we allow four feeding points and drive them alternately, all edges can be made to light up. The table below shows which circuits are activated by applying a voltage to the vertices. Since vertices 0/1 feed four current paths in parallel, we need either twice the current (relative to 2/3) or twice the on-time.

=== Driving Scheme ===
Path 1: 0 -> 2 -> 1 | 0=A 1=C
Path 2: 0 -> 3 -> 1 | 0=A 1=C
Path 3: 0 -> 4 -> 1 | 0=A 1=C
Path 4: 0 -> 5 -> 1 | 0=A 1=C
Path 5: 2 -> 4 -> 3 | 2=A 3=C
Path 6: 2 -> 5 -> 3 | 2=A 3=C

Bipolar Driving with 2 feeding points

Interestingly, there is another solution that allows driving all edges with only two feeding points: Bipolar driving, where we apply an alternating voltage.

The images above show how this approach works. The path length is \(L=3\), and the two feeding points are connected to opposite vertices of the octahedron. The table below shows the paths that are activated during each cycle of the alternating current driving scheme.

=== Driving Scheme ===
Path 1a: 0 -> 5 -> 3 -> 1 | 0=A 1=C
Path 1b: 0 -> 5 -> 2 -> 1 | 0=A 1=C
Path 2a: 0 -> 4 -> 2 -> 1 | 0=A 1=C
Path 2b: 0 -> 4 -> 3 -> 1 | 0=A 1=C
Path 3a: 1 -> 5 -> 2 -> 0 | 1=A 0=C
Path 3b: 1 -> 5 -> 3 -> 0 | 1=A 0=C
Path 4a: 1 -> 4 -> 2 -> 0 | 1=A 0=C
Path 4b: 1 -> 4 -> 3 -> 0 | 1=A 0=C

The four edges in the middle of the path (5->3,5->2,4->2,4->3) are part of both the forward and backward biased directions - the filaments leading to them act as a full-bridge rectifier. Since the current for the middle segments is branched (2x), the middle segments receive half the current. But since they are driven in both directions, the time-averaged current is the same as for the outer segments.

Driver Board

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How do we implement this in hardware? I designed a driver board based on a CH32V003 MCU and multiple H-bridge motor-driver ICs, which can drive up to 12 feed points with configurable anode, cathode, or high-Z states at voltages up to (and beyond) 10 V and cycle through arbitrary driving schemes at a few kHz. More details here. Below you can see the driver board next to an octahedron illuminated with bipolar driving.

Identifying other shapes

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The examples above were found through intuition and manual exploration. But what about more complex objects? Which ones are eligible, and how should we orient their edges to optimize the driving scheme?

A polyhedron (plural: polyhedra or polyhedrons) is a three-dimensional solid with flat polygonal faces and straight edges. The simplest examples are the five Platonic solids: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Beyond these, there are hundreds of polyhedra that have been catalogued by mathematicians. The Polyhedra Viewer by Nat Alison provides a nice interactive catalog, and is also a good source of 3D object files.

For our purposes, each polyhedron defines a filament object: vertices become electrical junctions, edges become LED filaments.

Finding solutions for any shape - randomized search

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Initially, I “vibe coded” a visualizer and solver in Python to find solutions for a given object under constraints, and visualize them. You can find it here.

However, I had to learn that a randomized search is not sufficient to find good solutions for larger objects due to an enormous explosion of the search space. Given a polyhedron with \(E\) edges and \(V\) vertices, there are \(2^E\) possible ways to orient the edges (assigning anode/cathode direction to each filament). For each orientation, there are \(3^V\) possible feeding point combinations (each vertex can be anode, cathode, or floating). This quickly becomes infeasible, even a simple cube has \(2^{12} \cdot 3^8 \approx 27\) million possibilities.

Instead, it is necessary to reduce the search space by limiting the number of options looked at. I found the most feasible approach was to assume certain driving schemes, as identified in the manual search above, and try to identify eligible objects for it.

Based on the exploration of the Octahedron and Cube above, we can observe three different configurations of filaments that allow driving an entire polyhedron with only few feeding points:

1. DC from two feeding points, ignoring brightness variation (as seen in the cube)
2. Multiplexing several DC paths to achieve constant brightness (as seen in the octahedron)
3. Bipolar driving of two feeding points with constant brightness (octahedron)

How do we screen for objects that support these driving schemes?

Case 1: DC driving from two points, Edge-Geodesic Cover

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Just to recap: We want to connect a positive terminal (anode) to one vertex and a negative terminal (cathode) to another vertex, so that all filaments (edges) lie on the current path between these two points. For all filaments to light up, every edge must lie on a shortest path between the two feeding points. We then try to identify such pairs of vertices, so that the shortest paths between them cover all edges. Note that this will result in a solution that allows all filaments to light up, but since there can (and will) be branching paths, the current distribution will be uneven.

This (and it took me quite a while to figure this out) is known in graph theory as the edge-geodesic cover problem. Specifically, we want to screen polyhedral graphs for the existence of an edge-geodetic set with an edge geodetic number of 2. This means that there exist two vertices (s, t) such that every edge in the graph lies on some shortest path between s and t. A necessary precondition for this to be possible is that the graph is bipartite, which can be easily tested for. Essentially, this states that the graph can be separated layer-by-layer without any dangling edges (filaments) between the layers.

Only 7 out of the 122 polyhedra I analyzed support this property and allow driving all filaments from only two connections. They are listed in the table below.

I built two of those objects: The hexagonal prism and the truncated octahedron. Below you can see photos of both objects in all their glory.

However, all objects allow solutions where the filaments can be covered by shortest paths between multiple combinations of feeding point pairs. This allows for DC solutions with multiplexing. I found that many objects require 4 taps, while the maximum I found was 8. You can explore further geodesic cover solutions in the interactive web viewer.

Case 2: Eulerian Circuits and Cycle Decomposition

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While the previous approach allows finding solutions that can light up all filaments, the brightness is usually uneven due to branching current paths. To achieve uniform brightness, we need a different strategy. One approach I found was to seek objects that can be decomposed into circular loops (cycles).

This is related to the famous Seven Bridges of Königsberg problem that Euler solved in 1736 and is usually cited as the birth of graph theory. An Eulerian path visits every edge exactly once, but only exists if the graph has exactly 0 or 2 odd-degree vertices. Most polyhedra fail this condition - the cube, for instance, has degree 3 at all vertices.

Instead of one path covering all edges, we look for a set of cyclic paths that together cover all edges exactly once. For uniform brightness, these cycles must all have the same length \(L\). The number of feeding point pairs needed is then \(m = \frac{E}{L}\). A necessary condition is that all vertices have even degree (typically 4), and the cycle length should equal the polyhedron’s graph diameter to prevent short circuits.

These conditions can be easily tested for. Only 5 polyhedra support cycle decomposition with uniform brightness:

Below you can see the cuboctahedron based on the cycle decomposition approach.

In addition, there are also some antiprisms that allow decomposition into cycles of equal length. However, in these the subcycles overlap, which leads to uneven brightness. Still, they are interesting shapes to build. The hexagonal antiprism decomposes into 3 cycles of length 8, which are color coded in the object below.

Case 3: Bipolar Driving

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As described above, it was possible to drive all edges of the octahedron at constant brightness with only two feeding points using bipolar (AC) driving. Can we generalize this to other shapes? I implemented a specific search algorithm to identify objects that support this driving scheme. The constraints are:

• At least 4 parallel paths between the two feeding points
• The “equator” vertices (middle layer) must have even degree
• No “diagonal” edges that would create shortcuts between layers

These constraints turn out to be quite restrictive. Only three polyhedra (plus one star-like variant) support bidirectional driving:

The elongated square bipyramid is basically an extended octahedron. In principle it would be possible to extend it further to create even longer objects.

The star octahedron is not really a polyhedron. It’s based on an octahedron where 4 edges are extended outwards to create a star shape. You can see it in all its glory below - my favorite star-like object!

Summary

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This concludes my exploration of glowing polyhedrons - a seemingly simple idea that consumed way more brain cycles than I had anticipated. I hope someone finds a better way to implement a general solver; my current approaches still appear unsatisfactory.

Should you feel inspired to build your own glowing polyhedrons, you can find all my exploration results, including filament characterization, driver board design, and the solver code in the GlowPoly repository.

I also created an interactive web viewer that allows you to explore all solutions I found for the 122 polyhedra in the database. You can find it here.