Friday puzzle -- visualize this

Part 1 -- visualize this

For part one you make a W shaped cut.

Lee

Re: Part 1 -- visualize this

Can't picture the W shaped cut solution. The one I was looking for is a straight cut. The intersection of a plane with a cube. Think of one pass through a table saw. (But please don't try this with a sugar cube!)

Re: Part 1 -- visualize this

It looks like this!

2 Important Questions

1. Is the coffee caf. or decaf.?

2. Is the donut glazed or cake?

Thanks for any clarification.

Tom

Worth(pic)

Sorry, I just don't see your W shaped cut. It just looks like a view of a cube to me. I don't see any cuts in it.

Maybe if you described it? If the cube has front, back, top, bottom, left, and right faces, with corresponding names for edges and vertices where those faces meet, where do you start your cut, and in what direction? Where do you change direction for the next three parts of the "W"?

Title

of that last post was supposed to read "Worth (pic) < Worth (1000 words)"

Guess it didn't like the "<" in the title.

You decide. The cube is cane sugar. ;-)

Another view

Here's another view.

Lee

Re: Another view

Looks like a SU file. Email it to me if you don't mind; maybe if I rotate it around a little, I will be able to see it. I just sent you a SU file of the plane cut solution.

Re: Worth(pic)

Descriptions aren't always easy. If I were to put one point up to the headstock, and the furthest point to the tail, when parted in the middle I'd be cutting on all 6 facets.

Yes (Part 1)

That is correct. For non-turners, corresponding to mounting the cube at the top-back-right corner and the front-left bottom corner, the hexagon is lines connecting the midpoints of the top-front, top-left, left-back, bock-bottom, bottom-right, and right-front edges. A picture of the cut (one cube with the cut lines, and the other with one of the cut pieces removed) is below.

Now, on to part 2, which is (for me) much harder to visualize.

Re: Part 1 -- visualize this

Lee: Got your file. Thanks. Once I rotated it a bit, I now can't figure out why I couldn't see it in your second pic!

I'd describe your cut as one of many facets, that when looked at from one angle appears to be a hexagon. Not quite the same thing.

Re: Friday puzzle -- visualize this

The first one was not too tough. After carefully lining things up in the compound miter saw, I got .707cm sides (and a lot of powdered sugar!)

The second one is harder to visualize. But it is related to your race-track puzzle. I believe a 1" dowel would make a good saw guide.

I don't think so

.707cm sounds about right (1/sqrt(2)) for part 1, but I don't see how you could use a 1" dowel for the cut that needs to be made in the donut. But maybe I'm missing the hint at an answer. What size(s) are the circles?

Re: I don't think so

You are right. Forget the dowel. My degenerate solution doesn't extend to the general case.

By process of elimination, I think that the diameters would have to be 2. But no proof of circularity (not to be confused with a circular proof) comes to mind yet.

Re: I don't think so

But even without the proof, can you describe the cut? And what process of elimination left you with two same-size circles of 2"? I'm not saying you are wrong ;-) just wondered how you ruled out 1 2" and 1 1" circle, or two 1.5" circles?

Re: I don't think so

Cutting vertically, you can get 1" circular sections, but they will be disjoint.

Cutting horizontally, you can get circles ranging from 1" to 3", but they will always be concentric.

The only other option is a diagonal cut. If you start at the top on one side and end at the bottom on the opposite side, you would pass through the center of the torus. Assuming that the curves produced are circular (i.e. trusting that you didn't lead us on a merry chase), the diameter would have to be 2" (from one outer edge to the opposite inner edge at the central horizontal plane)

Getting close

Cutting vertically, you can get 1" circular sections, but they will be disjoint.

Cutting horizontally, you can get circles ranging from 1" to 3", but they will always be concentric.

The only other option is a diagonal cut.

[b]So far, so good.[/b]

If you start at the top on one side and end at the bottom on the opposite side, you would pass through the center of the torus.



And you would be cutting a pretty close to the correct angle.

Assuming that the curves produced are circular (i.e. trusting that you didn't lead us on a merry chase),

[b]But those curves aren't circular. The other possibility is that I didn't lead you astray, but that another slanted cut is what is called for. ;-)[/b]

the diameter would have to be 2" (from one outer edge to the opposite inner edge at the central horizontal plane)

[b]Why would you measure from the outer edge of one side of the torus when your cut is starting at the high point of the torus? And why measure along the horizontal plane, when the circles in the faces are on the slant?

[/b]

Answer, Part 2

Bill was VERY close to the correct cut. The slanted cut is the one that is tangent to each side of the torus, which in the case of a torus with a hole the same size as the tube is 30 degrees from horizontal. The image below shows several views of both the donut with the plane of the cut and one half of the cut donut.

I still can't see it without the crutch of Sketchup.

My understanding is that it can be proven that this is the only slanted cut that produces intersecting circles, but I haven't seen, much less duplicated, such a proof.

Re: Answer, Part 2

And yes, Bill, they are 2" circles.

Re: Getting close

[b]But those curves aren't circular. The other possibility is that I didn't lead you astray, but that another slanted cut is what is called for. ;-)[/b]

My description wasn't quite precise enough. I was in a rush this morning to get out the door for some spring-skiing.

The cut does start at the top on one side and and end at the bottom on the other. However, in order for the curves to cross the plane must be tangent to the surface of the torus at two points along the center-line of the cut.

If you were to draw a line through the center point of the torus and tilt it until it contacted the surface on both sides of the 'donut hole', that would be the angle of cut.

[b]Why would you measure from the outer edge of one side of the torus when your cut is starting at the high point of the torus? And why measure along the horizontal plane, when the circles in the faces are on the slant?

Because we are assuming that the curves are circles and circles are - um - circular. If it a circle with a center on the horizontal plane and it is two inches wide as it crosses the horizontal plane, than it is two inches in diameter.

Re: Answer, Part 2

Another excellent puzzle Alex.

Thanks!

Re: Getting close

AY: Why would you measure from the outer edge of one side of the torus when your cut is starting at the high point of the torus? And why measure along the horizontal plane, when the circles in the faces are on the slant?

BE: Because we are assuming that the curves are circles and circles are - um - circular. If it a circle with a center on the horizontal plane and it is two inches wide as it crosses the horizontal plane, than it is two inches in diameter.

Doh!!

Re: Answer, Part 2

Credit where credit is due: This was a Martin Gardner puzzle.