Tangent Circles: Newsletter Challenge (07/24/07)

Bucky-balls have more faces than can be achieved in this way. (You can only have the five regular solids) But bucky-balls can make better approximations to spheres than the regular solids because of the higher number of faces. However, the circles in the pentagons would inevitably be smaller than those in the hexagons, given that the edges are shared.

N.B that the basic carbon versions are modifications of the graphite sheet, so they inevitably have edges of unequal length, so there is no proper "inscribed" circle.

Sorry, I should have been more specific.

A sphere with its surface covered with tangent circles of number n and equal radius r.

I think that gives just 5 possibilities for n: 4, 6, 8, 12, and 20

Fyz

Clarification:

I should have said that you can draw an arbitrary number of circles on the surphace of a sphere, which share tangents with up to five other circles*; but that these are the only arrangements that give uniform coverage.

*However, if all circles must share tangents with the same number of other circles, the above five remain the only possibilities

Seven equal circles of radius r can be fitted into a large circle of radius R=3r. This is done by constructing a regular hexagon of sides 2r, the 6 circles on the vertices's of the polygon and the 7^th at its center (same center of the large circle).

Now, consider the radius R same for the sphere.
Surface area of the sphere, A=4лR² is 4 times area of the large circle, лR².

Can we then deduce that 28 tangent circles of equal radius r can cover the surface of the sphere of radius R=3r?

Not directly, if at all (unless you allow some of the circles to overlap - in which case you can have as many as you care to draw). The issue is that you can surround a circle with six circles of the same size only if you are drawing on a flat surface. On the surface of a sphere, the maximum equal-size surround is five, so the gaps are likely to be larger. In addition, the surface area of the sphere that is "enclosed" by a circle is greater than ∏r².

N.B. Of course, if you want the five circles to make a continuous ring, that makes the sphere the right size to have a total of 12 tangent circles on its surface.

Fyz

I haven't heard of this acronym, but I have heard of two others. One is for the order of solving equations, the other is for values of tangent at 0°, 90°, etc.

For equations, it's PEMDAS - Parentheses, Exponentials, Multiplication, Division, Addition and Subtraction. Or, "Please Excuse My Dear Aunt Sally."

For tangent values it's the OHIO IOU. Convert the O's to 0's and the I's to 1's, ingnore the H and the U.

Thinking outside the square again here. Would this qualify as a sphere of tangent circles?

I only got AutoCAD LT at work. This example is a bit like a manual wireframe. To do this I first drew a circle in normal xy plane, then rotate the UCS about the x axis by a set angle (360/'n') and then draw the next circle from the common center point (using osnap) using 'r' as the radius.

Hi Andyman,

That's AMAZING!

Please refer to my post #59 for clarification.

Thanks

The drawing by Andyman has all the circles sharing 2 common intersections. Looking at your #59 again, it reads (to me anyway) as if there can be any number of intersection nodes. To put it another way, like a sphere with randomly orientated circles drawn around it*. The description you give does not define the shape that Andyman has drawn.Hope you see what I'm trying to say, since I have zero chance of drawing it.

* Last minute thought. Yours could look like a tennis ball with lots of elastic bands stretched around it (sort of).

I understand what your saying Kris. Even though all my circles are evenly spaced and rotated about 1 axis, the circles could rotate around any axis that bisects the center of the 'sphere' and all the circles would still be tangent.

Phew , glad it made sense. With an infinite number of circles either case would still produce the surface of a sphere. Nit-picking the fine detail is half the fun of these Challenge Questions. I'm going to have to bite-the-bullet and play around with some CAD (3Doug mentioned having another similar problem in mind).

I don't understand your statement "all the circles would still be tangent". These circles intersect at a point. A pair of tangent circles shares a common tangent - the directions of the curves are identical at the point where they touch.

Fyz

Hi Willy,

I can see all the platonic solids as being achievable. Take my 1st example of 3 tangent circles; those 3 circles could be inscribed inside a triangle which would then be used to make a Icosahederon. Im not sure if I want to try drawing that though with AutoCAD LT

Got me thinking though, what about Spaceship Earth? Geodesic dome equations may have an answer (although I thought that the angles are not always perfectly equal)

SFIK, most geodesics would not strictly fit exactly on a sphere, and many use irregular polygons as well. Those that achieve both (like the football) would not result in all equal size circles.

Just for grins, I decided to turn Randall's ilustration from post #12 into a solid.

I gave it the color I did because I thought it looked more like a piece of candy than a donut. Now that I mentioned that, aren't you getting hungry?

Maybe I'm showing off a little bit here. Maybe I'm just a bit fruity....

That's quite nice and relaxing. It's like looking at a pregnant alien through a telescope. Or maybe some of Princes Blood.

No, It's just been a long night, I don't do drugs. So called 'British Racing Green' would have been nicer, but it's still good. Can you apply a wire frame to it, in order to show contour better ?

Would it look more like a doughnut if you iced it and sprinkled it with sugar? Perhaps a bit if strawberry jam oozing out of one side... (that's the closest I could find to licking my lips)

I would have made it look like a chocolate cake doughnut, but Inventor doesn't have that in the As Material list. It does have grape and orange, so I went with grape.

I prefer cream-filled doughnuts. If I want jelly, I'll make a PBJ.

Also, a good test of doughnut shops, in my experience, is blueberry. If a shop makes good blueberry doughnuts, everything they make will be good!

Now I'm making myself hungry! I've already had supper, but I still have time for a snack tonight!

I have a strange sense of deja vu.

As you'll see, I have gone back to designing with paper. And yes, I again lack focus.

I'll start with 4...that is one of those children's counting game tihings, isn't it?

Nah, 6-pointed star made from a square of paper (a 'Jackstone' yes). I use bones for the game. Got bored while my computer was running chores on auto pilot. Nice relaxing therapy to stop me dribbling and drooling !

Well, here I go again. This looks more like a grape, but probably one from another planet!

I made this in Inventor, like I did the previous one. This time I revolved the profile instead of extruding it. I started with the same ring of circles I used for the sprocket, except I trimmed the insides of the circles, and added a vertical line for the axis. At first, it didn't want to revolve, so I deleted the circles on one side of the vertical line, and it worked.

"So, you say you want a revolution?"

OOOPS! I did revolve the previous one. I must have been thinking of the one previous to the previous one!

I must have missed whatever this was responding to, because this looks like a sphere - not the doughnut needed for tangent circles. Please enlighten.

Fyz

The first solid I produced, the sprocket, I made by extruding. I made the second one, the grape donut-shaped candy, by revolving a circle around a vertical line tangent to the circle. The third is a "sphere" made by revolving half of the ring of 50 circles used for the sprocket, except I trimmed the circles on the inside, and added a vertical line to close the profile and serve as the axis of revolution.

Sorry about the confusion. Is this clear as mud now?

Thanks - clear as a bilberry now. (Although perhaps you get an infinite number of pairs of tangent circles where two tangent circles where two bilberries touch?)

Perhaps I'm getting(?) obsessive about staying on thread - maybe spending too much time with my tailor?

Fyz

If using drafting software such as AutoCAD, I would draw a circle with R radius and use the "array" command to draw N number of circles based on a row and/or column spacing of R radius and N being equal to the number of rows multiplied by the number of columns.

How would that give a ring of tangent circles? Or didn't you read post #28?

Kinda ironic that the logo for Mopar was a pentagon, a regular polygon with n = 5 sides....

When was that? Can you provide a link? (Assuming you don't mean the star)

The classic logo for Dodge cars is the 5-pointed star inside a pentagon. Chrysler used this logo also when it distriubted OEM parts under the Mopar label. Dodge and Plymouth race cars and high-performance engines were often called Mopars.

You said " using CAD software" so I exspected a drawing not a written expanation.

Please show us the drawing.

3E

see post 75 where the intended problem was clarified and where 3Doug also showed the final drawing of a chain sprocket that was developed using the solution

originally clarified in post 28;you won't be able to see any difference between the drawing shown in 51 and one produced by this method

drafting software has a command "array" where you spec type of array and number of elements. just draw the circle and "array" it.

Yes, that's one interpretation of the problem as presented. Close-packed circles would require a little more care in specification.

However, the intended problem was a ring of circles - as explained in post 28.

Here's an illustration of the relationship between a regular polygon of n sides and a ring of n tangent circles:

Exekiel49 was the first to pick up on this.

Andyman came up with a very good alternative method in AutoCAD.

AutoCAD has a Polygon command that creates a regular polygon. The command first asks for the number of sides, then you pick a point for the center of the polygon. Next it will ask if the polygon is to be inscribed inside a circle or circumscribed around a circle. Finally, it asks for the radius of the circle.
Using this command requires that you know the radius before hand. Because this requires doing some trig, I avoided that in formulating the official answer. Also, not every other drafting program has a similar command. You can use the method in the official answer with any drafting program, or even pencil (or pen, if you're THAT good!) and paper.
I must commend Kris for providing the math needed to find the radius of the circle in which the polygon is to be inscribed if using the AutoCAD Polygon command. The radius is the same as the hypotenuse of a right triangle formed by a line from the midpoint of one side of the polygon to the center point of the polygon, and a line from the center point to a vertex, and the half of the polygon side between these lines. Here's an illustration of this relationship:

The formula works out to be hyp = sin (180°/n)/r. I used this formula to create these two illustrations with n = 10 and r = .5.

As an interesting side note, for the spherical grape-colored shape I created in Inventor (post #119), I first created a 50-sided polygon using the polygon command, then created a dimension with a length of 2r, and the program automatically resized all sides to that length.

Now, for Ezekiel, Andyman and Kris, if you wish a reward, here it is:

Nice summary. I'm pretty sure that anybody here could have done the calculation bit. It's cool that you followed up with a nice CAD explanation + the pictures as well. We don't' often get to see nice answers like this on a Challenge Question. It all adds to the variety of fun/learning etc. < Applause for 3Doug>

"The formula works out to be hyp = sin (180°/n)/r"

If hyp is hypotenuse, shouldn't that read "hyp = r / sin (180°/n)"?

Fyz

'hyp' is the very cool 'inverse-hypotenuse function. Maybe.

Fyz,

You are right! While manipulating the formula, I must have left out the slash in

sin (180°/n) = r / hyp.

Thanks for catching that. The strange thing is the drawing still shows the relationships correctly, it's just the sizes didn't match.

I guess you deserve a reward too, so here it is: