# A pen is attached to the wheel. Assume the pen is a sphere
**URL:** https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457
**Category:** Factual Questions
**Created:** [August 25, 2025, 2:10pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457 "2025-08-25T14:10:21Z")
**Posts on this page:** 20
**Page:** 1
### Author:  [@OldOlds](https://boards.straightdope.com/u/OldOlds)
#### Post date: [August 25, 2025, 2:10pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/1 "2025-08-25T14:10:22Z")
OK. I don’t know how I’m going to explain this.
Observing a bicycle with a reflector on the wheel drive by the other night, I was looking at the reflector go round and round. And I wondered: If that reflector had a pen that drew a line on a piece of paper beside the bicycle, what would the resultant “wave” look like? Would it be some sort of modified sine wave? Would the line cross over itself at the bottom when the pen goes from moving in the direction of the bike to away from it? How much would the distance from the center of the wheel affect things?
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### Author:  [@iiandyiiii](https://boards.straightdope.com/u/iiandyiiii)
#### Post date: [August 25, 2025, 2:14pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/2 "2025-08-25T14:14:08Z")
Depends on the size of the wheel, right?
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### Author:  [@Kron](https://boards.straightdope.com/u/Kron)
#### Post date: [August 25, 2025, 2:14pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/3 "2025-08-25T14:14:18Z")
I believe it should just look like a simple scalloped edge line.
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### Author:  [@Marvin\_the\_Martian](https://boards.straightdope.com/u/Marvin_the_Martian)
#### Post date: [August 25, 2025, 2:15pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/4 "2025-08-25T14:15:36Z")
If the pen in on the rim it would draw a sine wave. If at the hub a straight line. In between a series of loops (sort of like a cursive “l” or “el) whose amplitude diminishes as the pen is moved from the rim to the hub.
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### Author:  [@susan](https://boards.straightdope.com/u/susan)
#### Post date: [August 25, 2025, 2:17pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/5 "2025-08-25T14:17:43Z")
Surely there’s a Spirograph component that would illustrate this.
What do you mean by “the pen is a sphere?”
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### Author:  [@RitterSport](https://boards.straightdope.com/u/RitterSport)
#### Post date: [August 25, 2025, 2:18pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/6 "2025-08-25T14:18:15Z")
On the rim, it seems to draw a cycloid. Inside the rim, it’s a curtate trochoid, assuming this is what you mean:
> **[Trochoid](https://en.wikipedia.org/wiki/Trochoid)**
>
> In geometry, a trochoid (from Greek trochos 'wheel') is a roulette curve formed by a circle rolling along a line. It is the curve traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle) as it rolls along a straight line. If the point is on the circle, the trochoid is called common (also known as a cycloid); if the point is inside the circle, the trochoid is curtate; and if the point is outside the circle, the trochoid is prolate. The word "trochoid" As...
> [@susan](#):
>
> Surely there’s a Spirograph component that would illustrate this
Spirograph was the first thing I though of, but the various wheels stay within the page, in a circle, rather than moving along the road.
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### Author:  [@Kron](https://boards.straightdope.com/u/Kron)
#### Post date: [August 25, 2025, 2:19pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/7 "2025-08-25T14:19:27Z")
Oh, that’s the word…. I knew there was a better word than “scallop”…
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### Author:  [@OldOlds](https://boards.straightdope.com/u/OldOlds)
#### Post date: [August 25, 2025, 2:20pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/8 "2025-08-25T14:20:19Z")
Wow. Some quick responses. I’m not surprised it has a name.
> [@susan](#):
>
> What do you mean by “the pen is a sphere?”
> **[Spherical cow](https://en.wikipedia.org/wiki/Spherical_cow)**
>
> The spherical cow is a humorous metaphor for highly simplified scientific models of complex phenomena. Originating in theoretical physics, the metaphor refers to some scientific tendencies to develop toy models that reduce a problem to the simplest form imaginable, making calculations more feasible, even if the simplification hinders the model's application to reality.
> The phrase comes from a joke that spoofs the simplifying assumptions sometimes used in theoretical physics.
> Milk production at...
Just a little joke
ETA: Though the Wiki version isn’t (IMO) as good as the one I herard. In the version I heard the punchline is "The physicist presents his results. ‘First, we assume the cow is a sphere…’” and lets the listener/reader infer the rest.
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### Author:  [@Munch](https://boards.straightdope.com/u/Munch)
#### Post date: [August 25, 2025, 2:21pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/9 "2025-08-25T14:21:19Z")
an animation:
[ ](https://www.youtube.com/watch?v=FqJ1RjJLjEs)
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### Author:  [@Elmer\_J.Fudd](https://boards.straightdope.com/u/Elmer_J.Fudd)
#### Post date: [August 25, 2025, 2:21pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/10 "2025-08-25T14:21:30Z")
Something like this?
[ ](https://upload.wikimedia.org/wikipedia/commons/6/69/Cycloid_f.gif)
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### Author:  [@Reply](https://boards.straightdope.com/u/Reply)
#### Post date: [August 25, 2025, 2:26pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/11 "2025-08-25T14:26:28Z")
Kinda related… if you make the “pen” out of an LED light strip and take a long exposure, you get something like this:
[ ](https://cdn-learn.adafruit.com/assets/assets/000/026/747/large1024/projects_Bike_Wheel_POV_Display_adafruit_01.jpg)
From a persistence of vision bike light project: [Overview | Bike Wheel POV Display | Adafruit Learning System](https://learn.adafruit.com/bike-wheel-pov-display/overview) (edit: fixed link)
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### Author:  [@Telemark](https://boards.straightdope.com/u/Telemark)
#### Post date: [August 25, 2025, 2:28pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/12 "2025-08-25T14:28:55Z")
Here’s a spirograph simulator - [Spirograph pattern generator - ScienceDemos.org.uk](https://sciencedemos.org.uk/spirograph.php)
You can try out some other options.
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### Author:  [@Hypno-Toad](https://boards.straightdope.com/u/Hypno-Toad)
#### Post date: [August 25, 2025, 2:47pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/13 "2025-08-25T14:47:44Z")
Imagine how much fun Euclid would have had with a spirograph.
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### Author:  [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)
#### Post date: [August 25, 2025, 2:49pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/14 "2025-08-25T14:49:40Z")
> [@RitterSport](#):
>
> Spirograph was the first thing I though of, but the various wheels stay within the page, in a circle, rather than moving along the road.
My recollection of Spirograph kits was that in addition to the hollow gears with teeth on the inside perimeter, the kits also included a couple of straight-line gear racks totaling maybe 12 or 18” of length.
So you could tack that straight track to your paper and use one of the convex circular gears with the teeth on the outside perimeter to roll along it. Which would be a strong analog to the bicycle wheel experiment. The smaller the gear you chose, the more revolutions you could get out of your e.g. 18” run.
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### Author:  [@RitterSport](https://boards.straightdope.com/u/RitterSport)
#### Post date: [August 25, 2025, 2:50pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/15 "2025-08-25T14:50:24Z")
I don’t think I ever had, or at least never used, the straight-line spirograph thing.
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### Author:  [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)
#### Post date: [August 25, 2025, 2:52pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/16 "2025-08-25T14:52:22Z")
I expect that over time they got more elaborate. And most likely started out with a basic kit and a “deluxe” kit with more gizmos.
So depending on which year you got yours, etc., you may never have seen the gear racks.
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### Author:  [@RitterSport](https://boards.straightdope.com/u/RitterSport)
#### Post date: [August 25, 2025, 2:53pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/17 "2025-08-25T14:53:58Z")
Probably in the early 70s? Anyway, that animator is really cool.
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### Author:  [@CookingWithGas](https://boards.straightdope.com/u/CookingWithGas)
#### Post date: [August 25, 2025, 2:56pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/18 "2025-08-25T14:56:20Z")
> [@Marvin\_the\_Martian](#):
>
> If the pen in on the rim it would draw a sine wave.
Closer to ABS(SIN)
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### Author:  [@crowmanyclouds](https://boards.straightdope.com/u/crowmanyclouds)
#### Post date: [August 25, 2025, 2:57pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/19 "2025-08-25T14:57:00Z")
> [@LSLGuy](#):
>
> I expect that over time they got more elaborate. And most likely started out with a basic kit and a “deluxe” kit with more gizmos.
Yup,
[ ](https://www.mandisattictoys.com/cdn/shop/products/IMG_5156_591ac663-7335-4502-b98d-576c8583150a_1024x1024.jpg?v=1620161242)
[ ](https://fredaf.wordpress.com/wp-content/uploads/2014/08/super-spirograph.jpg)
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### Author:  [@Reply](https://boards.straightdope.com/u/Reply)
#### Post date: [August 25, 2025, 2:59pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/20 "2025-08-25T14:59:55Z")
By the way, they recently relaunched the Spirograph toys at Target: [Spirograph products at Target](https://www.target.com/b/spirograph/-/N-55d1o) (along with a whole host of Gen-X/Millennial nostalgia toys). There’s an [endcap display that takes you right back to the 80s](https://www.seanpcarlin.com/figure-ground-reversal/target-retro-endcap/):
[ ](https://i0.wp.com/www.seanpcarlin.com/wp-content/uploads/2019/12/Target-retro-endcap-scaled-e1576795113590.jpg?fit=676%2C901&ssl=1)
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