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Icosagon top
The good and bad news: The top's 20-sided red ring has symmetries rarely encountered in the LEGO realm.
About this creation
Please feel free to look over the images and skip the verbiage.

The spinner from my Emotigon feeling picker easily converts to a pretty cool finger top.





What intrigues me most about this top is the clash of symmetries it embodies:
  • The symmetries of the red ring, which are rarely encountered in the LEGOŽ realm.
  • The symmetries usually attainable with LEGOŽ construction, and the limitations they placed on the chassis supporting the ring.
  • The mass symmetries needed for any top to stay up and spin smoothly.
Those who know my work will be shocked to hear that the top actually came before the Emotigon and its spinner.



In fact, the "-gon" in "Emotigon" came from the ring, which has the shape of a simple concave equilateral icosagon (20-sided polygon). I do my best to bore the reader with more on this interesting geometry below.

But enough about the Emotigon. This page is strictly about the top. The photos will show its ring in 3 different color schemes, but their mass distributions are identical.







Visually, I prefer the middle one with the embedded black pointer. The multi-colored version is for gambling. Here's the plain one in action...



I can spin this top to 500-600 RPM by hand. The single-stage 1:4 planetary starter in the video can take it to ~700 RPM, but I was using the starter more for tilt control, as this top performs best with a vertical release not easily obtained by hand.



This second-generation 2-stage 1:16 planetary starter provides the same tilt control with release speeds of 800-900 RPM, but alas, the assocated spin time gains are meager due to the top's poor aerodynamics.

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Structure

This studless top is basically just a polygonal "ring" supported by a much lighter 2-spoke "chassis". The latter includes a "stem" for the fingers, a "tip" for the ground, and a "hub" for the spokes in between.



The stem and tip are "axial" structures in that they define the intended spin axis. Structures like the ring and spokes perpendicular to this axis are "transverse". The transverse "ring plane" containing the ring and spokes and passing through the ring's own center of mass (CM) serves as a handy spatial reference.



∨ Ring: The ring's clearly the visual star of the show, both at rest and at speed.



But the ring's more than just a pretty face. It also controls spin time and smoothness through its domination of the top's total mass, mass distribution, and aerodynamic drag.

The ring divides both structurally and functionally into "frame" and "ballast". The frame is the black part in this color-coded mock-up. The short red, white, and blue liftarms and their black and LBG pins make up the ballast.



The spokes of the chassis attach at the 2 bare LBG pins. Unfortunately, these were the only spoke-ring attachments possible with available hub geometries.

The frame itself is a just closed loop of 10 inward-pointing 1x7 bent liftarms. We'll call these "segments" for short.



Note that the frame has 2 concentric sets of vertices: The 10 "outer vertices" where frame segments meet, and the 10 "inner vertices" evenly spaced in between. Here, the outer vertices are centered on the red 2L axles, and the inner vertices on the pin holes at the liftarm bends.

The gray and blue color scheme in the last photo emphasizes some of the frame's symmetries and asymmetries. For starters, the unavoidable "staggering" of frame segments above and below the ring plane reduces the frame's mass rotational symmetry from 10-fold to 5-fold.

The good news: Five-fold symmetry is plenty. Two-fold symmetry in mass (not color) is enough to guarantee perfect static balance about the spin axis, and 3-fold is enough to keep the top from falling over right away.

The bad news: The staggering of 10 segments leaves the frame with an inherent couple unbalance, which arises when mass elements balance across the spin axis but not in the same transverse plane. Problem is, a top needs both static and couple balance to spin smoothly.

The fix: Cancel the couple unbalance of the frame using a ballast of equal and opposite couple unbalance (here in red, white, and blue) while taking care to preserve static balance and proper rotational symmetry.

Finding the right ballast arrangement took some fiddling.



Ultimately, the role of the ballast is to reduce wobble. It would do an even better job if it also stiffened the frame against torsion and centrifugal expansion, but the necessary parts just don't exist at this scale.

The frame doesn't quite close on its own, but that's OK: The closing strain adds some stiffness to the ring, and every little bit helps in the fight against wobble.



In the closed frame, the internal and external angles at the outer vertices respectively measure 91° and 89° rather than the usual 90°. Since the segment bends measure 127° before and after closure, the 2L axles must be taking up all of the frame's closing strain.



∨ Chassis: The simple 2-spoke chassis (i) supports the weight of the ring when the top is spinning, (ii) transmits torque from stem to ring during spin-up, and (iii) works with the ring to suppress wobble due to elastic vibration of the top's structure. It does a good job with (i) and (ii) but struggles with (iii) at certain speeds.



The chassis includes the stem, the tip, the spokes, and the beefed-up hub used to bind them all together as stiffly as possible. The spokes have 2 plies to reduce couple unbalance and suppress twisting and flapping motions during and after spin-up.

Given the symmetry requirements for tops already discussed, you may be wondering why there are only 2 spokes. Well, I've made many a top with spokes, and the only workable spoke symmetries generally available in the LEGOŽ realm are 2-, 3-, 4-, 6-, and rarely 12-fold.

Of these, the 2-fold option is the only one shared by workable spoke attachments to the ring -- hence, just 2 spokes.





Once the ring is closed, miracle of miracles, the spokes span it perfectly without further strain. Good thing, too, because this particular ring would never have made it into a LEGOŽ top otherwise.

Now my previous statement that a top needs at least 3-fold rotational symmetry in mass to stay up at all needs some qualification. In practice, it's enough to have most of the top's axial moment of inertia (AMI) residing in components with at least 3-fold mass symmetry. I got away with 2-fold spokes here only because they account for less than 5% of the total AMI. The rest of the top is at least 4-fold.



The hub joining the stem and tip to the spokes is basically a pair of coaxial Hero Factory weapon barrels (98585). This arrangement also makes for a fairly rigid 4-spoke hub I've used many times before.

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Optional: Idealized ring geometry

Together, the 10 inner and 10 outer vertices of the ring define an idealized icosagon (20-sided polygon) here taken to lie in the ring plane. This icosagon has some interesting geometric properties.

NB: To avoid confusion, I'll be using "icosagon" to refer to the ideal geometric figure and "ring" to denote the plastic approximation embedding it.

A polygon is basically a closed chain of straight line segments sharing a common plane. The line segments are called "edges", and the points where they meet are called "vertices". Following most authors, I take the polygon to include only the edges and vertices and not the "interior" they enclose.

If a polygon has n edges, it also has n vertices. In an icosagon, n = 20 by definition, but that leaves a lot of room for variation.

In this ring mock-up, the embedded icosagon's inner vertices are centered on the LBG pins, and the outer vertices on the red 2L axles. The icosagon's edges connect only adjacent vertices and are identical in length at 24 mm. None cross the interior.



This icosagon is "simple" in that it never intersects itself. It's also "concave" in that some of its diagonals lie outside it. (A "diagonal" here is just a line segment connecting 2 points on different edges of the same polygon. Like a chord of a circle, a diagonal doesn't have to pass through the polygon's center.)

The concavity here follows from the fact that the icosagon is "acyclic". In a "cyclic" polygon, all vertices lie on the same circle. In this case, however, the inner and outer vertices lie on concentric circles of 2 different radii (56 and 70 mm).

The idealized ring icosagon is "equilateral" in that all edges are identical in length (in this case, 24 mm). But it's not "equiangular", as the inner and outer vertices respectively point inward and outward and enclose different angles (233° and 91°). To be "regular" like a square, our icosagon would have to be both equilateral and equiangular.

The icosagon is "star-shaped" in that all of it can be seen from at least one interior point. In this case, the entire icosagon is visible from any point within the 10-sided "core decagon" defined by the inner vertices. This decagon is also simple and star-shaped, but unlike the icosagon, it's convex, cyclic, and regular.

Like its core decagon, the icosagon has perfect 10-fold rotational symmetry about the spin axis. As suggested by the spoke attachments, it also has 2-fold rotational symmetry about the spin axis and another 10 transverse axes, each passing through the center and 2 opposing vertices. There are also 10 axial mirror planes, each passing through one of these transverse axes.

These symmetries have an important consequence: If the icosagon's edges were made of uniform rods of the same mass, it would have perfect static balance and perfect couple balance about its geometric center.

As built, the top's ring also has static balance, but the ballast doesn't cancel the couple unbalance perfectly. The residual couple unbalance may not be enough to cause visible wobble on its own, but it could excite wobble-inducing vibrations of the not-so-stiff ring and spokes.

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Optional: Performance

The icosagon top is more a looker than a performer. Spin times are middling at best, and wobble, though generally tolerable, tends to build as spin-down progresses. Play value is decent, but the wobble bugs me.



The top's mass distribution is certainly conducive to long, smooth spins, but the ring and spokes generate a lot of aerodynamic drag and aren't as stiff as I'd like. The last accounts for most of the wobble.

With a mechanical aid like this second-generation 1:16 planetary starter, release speeds consistently fall in the 800-900 RPM range, and spin times in the 30-33 sec range. When twirled by hand, however, release speeds drop to 500-600 RPM, and the top stays up only 24-28 sec.



To maximize spin time with either method, you need a release tilt of close to 0°. (Tilt is the angle the stem makes with the vertical.) That's easy enough to arrange with the starter, but it takes a very lucky twirl by hand.

Wobble is usually minimal during early spin-down but then grows with time regardless of release speed or tilt. I think it's mostly a speed-dependent vibration of the ring and spokes, but a small residual couple unbalance in these structures could be the trigger. Stiffening the ring and spokes would probably eliminate most of this wobble, but the top's visual charm would be lost. Fortunately, static unbalance isn't an issue here.



I mentioned earlier that the ring and spokes generate a lot of aerodynamic drag. Some of it comes from the corrugated shape of the ring and from the smaller-scale surface roughness created by all the exposed ring and spoke pin holes. The rest comes mainly from the spokes' broadside motion through the air. Streamlined spokes would help a lot here, but the necessary parts don't exist.

The total drag manifests itself as a braking torque that grows rapidly with speed. Tip friction also generates a small braking torque, but it varies little with speed and generally pales in comparison. The total braking torque (TBT) acting on the spinning top is solely responsible for its spin decay and may limit release speed as well -- especially with the starter. Spin time suffers either way.



You can gauge the air flow stirred by the top with your hand. In general, the stronger the flow at some standard distance from the top, the greater the TBT and rate of spin decay.

Partially offsetting the aerodynamic toll on spin time is the top's favorable mass distribution, which works its magic here in 2 independent ways -- (i) by lowering the spin decay rate at all speeds despite the high TBT, and (ii) by lowering the "critical speed" below which the top loses its stability against gravity and will fall at the slightest provocation. These benefits flow directly from the top's low center of mass (CM) and from fact that most of its mass is concentrated in a ring far from the spin axis. I'll spare you the details.

Unlike most of my LEGOŽ tops, this one isn't self-righting. Let me explain. Below a certain "critical tilt", a spinning top with even a little tip friction will typically right itself, meaning that its average tilt will slowly decay to 0° and stay there for a while. Then average tilt will start growing again until the top falls.

Most of my tops are self-righting with critical tilts in the 5-15° range, but the best the icosagon top can do is to stay vertical for a while if released at exactly 0° tilt. Otherwise, tilt will only grow with time, as seen in the video at 1:09. I have no explanation for this behavior, but my Xenotron top does the same thing despite having little else in common.

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Top specifications

Overall dimensions:148x93 mm (DxH)
Mass:59.1 g
Outer ring radius:74 mm
Inner ring radius:60 mm
Ring mass share:71%
Ring AMI share:~95%
Best spin times:28 sec by hand, 33 sec with 1:16 planetary starter
Best release speeds:~600 RPM by hand, ~900 RPM with the starter
Modified LEGOŽ parts:Tip cut from 4L antenna
Non-LEGOŽ parts:None
Credits:Original MOC
See also:Emotigon, studless tops, and LEGO spinning top folder





Idealized icosagon geometry

Polygon type:Simple, concave, equilateral, star-shaped
Faces:1
Edges:20
Vertices:20
Euler characteristic:1
Edge length:24 mm
Vertex types:2 (outer and inner)
Outer vertex radius:70 mm
Inner vertex radius:56 mm
Interior angles:91° and 233°
Exterior angles:89° and -53°
Rotational symmetry:10-fold
Wikipedia pages:Icosagon, Polygon

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Comments

 I like it 
  August 11, 2017
Many of your creation would be matching for a LEGO dacta set to teach geometry and some elementary physical laws, my friend! And you're a very good teacher, too... ;) (No matter what Jonathan felt...) If you're able and in the mood to do your stuff in LDD, you could go to LEGOideas: I believe there is a big blank to fill up with a creative spinner/top/Jeremy- stuff set! :)
 I made it 
  August 9, 2017
Quoting Nick Barrett It's a striking looking thing, and maybe surprising that the angle of those bent liftarms wouldn't lend itself to say, a 12 section design... I know that 10 spoke wheels would be a nightmare to do! 12s are easy.
Thanks, Nick! Yes, I had your recently posted big wheel hub in mind when I listed 12-spoke hubs as possible in the LEGO realm. Problem is, it would take a lot of strain (bending of liftarms + twisting of connecting axles) to get a ring of even 11 of these liftarms to close with all the bends pointing inward -- in part, because the liftarms are constrained to join at nearly right angles. Granted, TLG's proprietary ABS has a very high elastic limit, but high enough to handle a closed ring of 12 liftarms? Only testing would tell.
 I like it 
  August 9, 2017
It's a striking looking thing, and maybe surprising that the angle of those bent liftarms wouldn't lend itself to say, a 12 section design... I know that 10 spoke wheels would be a nightmare to do! 12s are easy.
 I made it 
  August 2, 2017
Quoting Doug Hughes Complex and interesting indeed! Plus it looks very cool ;)
Many thanks, Doug! This one's rapidly becoming a visual favorite of mine, in part for the video effects.
 I like it 
  August 2, 2017
Complex and interesting indeed! Plus it looks very cool ;)
 I made it 
  August 2, 2017
Quoting Werewolff . Well, that was a read and a half! A great spinner, and the level of information that came with it was mind-boggling, to say the least. But great work, like always!
Very kind. I seem to revel in things that turn out to be a lot more complicated -- and therefore a lot more interesting -- than they look, and this top is a prime example.
 I like it 
  August 1, 2017
Well, that was a read and a half! A great spinner, and the level of information that came with it was mind-boggling, to say the least. But great work, like always!
 I made it 
  August 1, 2017
Quoting Jonathan Demers Great work as always Jeremy, though all the geometry made my head hurt! ~Jonathan
Thanks, Jonathan! Sorry about that. If you happen to like geometry, as I clearly do, this top has a lot to offer. If not, you can always opt out of the part that hurts.
 I like it 
  August 1, 2017
Great work as always Jeremy, though all the geometry made my head hurt! ~Jonathan
 
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LEGO models my own creation MOCpages toys shop Icosagon top


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