Re: [math-fun] What is the probability that a randomly thrown polyhedral die will land on a given face?
Good point!* I will now try to find if something comparable has been done for more general polyhedra. —Dan —————————————————————————————————————————————————————————————— * If you've never tried it, get a fat mass-market paperback (i.e., small format, c. 4"x7"), put a rubber band about it to hold it together, and note the disparate efforts needed in order to get it to spin 360º midair about the three axes perpendicular to the faces of this rectangular solid.** ** "Rectangular solid"??? Surely there is a shorter and punchier word for this common concept! Andy Latto <andy.latto@pobox.com> écrit: ----- Assuming the die has 3 distinct moments, there are two stable ways to have the die spinning as it is tossed. I don't see any reason to expect the probabilities will be the same for these two different ways to throw the die. Or the unstable ways, but those are harder for the thrower to reproduce accurately. -----
The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped". Some author, I can't remember who, writes "2-box" and "3-box" and the like. On Wed, Jan 30, 2019 at 6:07 PM Dan Asimov <dasimov@earthlink.net> wrote:
Good point!* I will now try to find if something comparable has been done for more general polyhedra.
—Dan
—————————————————————————————————————————————————————————————— * If you've never tried it, get a fat mass-market paperback (i.e., small format, c. 4"x7"), put a rubber band about it to hold it together, and note the disparate efforts needed in order to get it to spin 360º midair about the three axes perpendicular to the faces of this rectangular solid.**
** "Rectangular solid"??? Surely there is a shorter and punchier word for this common concept!
Andy Latto <andy.latto@pobox.com> écrit: ----- Assuming the die has 3 distinct moments, there are two stable ways to have the die spinning as it is tossed. I don't see any reason to expect the probabilities will be the same for these two different ways to throw the die. Or the unstable ways, but those are harder for the thrower to reproduce accurately. -----
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Actually, it's parallelepiped. On 30-Jan-19 18:19, Allan Wechsler wrote:
The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped". Some author, I can't remember who, writes "2-box" and "3-box" and the like.
On Wed, Jan 30, 2019 at 6:07 PM Dan Asimov <dasimov@earthlink.net> wrote:
Good point!* I will now try to find if something comparable has been done for more general polyhedra.
—Dan
—————————————————————————————————————————————————————————————— * If you've never tried it, get a fat mass-market paperback (i.e., small format, c. 4"x7"), put a rubber band about it to hold it together, and note the disparate efforts needed in order to get it to spin 360º midair about the three axes perpendicular to the faces of this rectangular solid.**
** "Rectangular solid"??? Surely there is a shorter and punchier word for this common concept!
Andy Latto <andy.latto@pobox.com> écrit: ----- Assuming the die has 3 distinct moments, there are two stable ways to have the die spinning as it is tossed. I don't see any reason to expect the probabilities will be the same for these two different ways to throw the die. Or the unstable ways, but those are harder for the thrower to reproduce accurately. -----
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<< The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped" >> I don't think so. See https://en.wikipedia.org/wiki/Parallelepiped I suggest "oblong" is the word required. WFL On 1/30/19, Mike Speciner <ms@alum.mit.edu> wrote:
Actually, it's parallelepiped.
On 30-Jan-19 18:19, Allan Wechsler wrote:
The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped". Some author, I can't remember who, writes "2-box" and "3-box" and the like.
On Wed, Jan 30, 2019 at 6:07 PM Dan Asimov <dasimov@earthlink.net> wrote:
Good point!* I will now try to find if something comparable has been done for more general polyhedra.
—Dan
—————————————————————————————————————————————————————————————— * If you've never tried it, get a fat mass-market paperback (i.e., small format, c. 4"x7"), put a rubber band about it to hold it together, and note the disparate efforts needed in order to get it to spin 360º midair about the three axes perpendicular to the faces of this rectangular solid.**
** "Rectangular solid"??? Surely there is a shorter and punchier word for this common concept!
Andy Latto <andy.latto@pobox.com> écrit: ----- Assuming the die has 3 distinct moments, there are two stable ways to have the die spinning as it is tossed. I don't see any reason to expect the probabilities will be the same for these two different ways to throw the die. Or the unstable ways, but those are harder for the thrower to reproduce accurately. -----
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As Allan says, there is always more physics. Somehow I can no longer find the web page about the d5 robot. Someone -- maybe it was Lou Zocchi, or Koplow Games? -- wanted to make fair 5-sided dice, using a prism with equilateral triangular base and whatever height made it all work out (which "must exist by continuity"). They built cut dice of a bunch of different heights at 1mm increments, and had a die-rolling robot with a computer vision system that could tell whether it landed on one of the triangular or rectangular faces. Turned out the correct prism height varied a lot depending on whether the die was being rolled onto a 5mm-thick or 15mm-thick piece of plexiglass. Sigh. I don't want dice that become unfair if I roll them onto a tabletop that is more bouncy or more sticky or more slanted. Or when I roll them underwater, or on the moon, or in a storm. Well, *maybe* I'll be OK with the d48 or d120, which are fair by symmetry except that some of the symmetries involve taking a mirror image. Those are fair until I want to roll them in a tornado, at which point a right-handed vs. left-handed funnel cloud would matter. https://www.mathartfun.com/thedicelab.com/d120.html --Michael On Wed, Jan 30, 2019 at 8:00 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped" >> I don't think so. See https://en.wikipedia.org/wiki/Parallelepiped
I suggest "oblong" is the word required.
WFL
On 1/30/19, Mike Speciner <ms@alum.mit.edu> wrote:
Actually, it's parallelepiped.
On 30-Jan-19 18:19, Allan Wechsler wrote:
The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped". Some author, I can't remember who, writes "2-box" and "3-box" and the like.
On Wed, Jan 30, 2019 at 6:07 PM Dan Asimov <dasimov@earthlink.net> wrote:
Good point!* I will now try to find if something comparable has been done for more general polyhedra.
—Dan
—————————————————————————————————————————————————————————————— * If you've never tried it, get a fat mass-market paperback (i.e., small format, c. 4"x7"), put a rubber band about it to hold it together, and note the disparate efforts needed in order to get it to spin 360º midair about the three axes perpendicular to the faces of this rectangular solid.**
** "Rectangular solid"??? Surely there is a shorter and punchier word for this common concept!
Andy Latto <andy.latto@pobox.com> écrit: ----- Assuming the die has 3 distinct moments, there are two stable ways to have the die spinning as it is tossed. I don't see any reason to expect the probabilities will be the same for these two different ways to throw the die. Or the unstable ways, but those are harder for the thrower to reproduce accurately. -----
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-- Forewarned is worth an octopus in the bush.
On Wed, Jan 30, 2019 at 8:00 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped" >> I don't think so. See https://en.wikipedia.org/wiki/Parallelepiped
Which is presumably why I was taught "rectangular parallelepiped" as a child, which was fun to say, but certainly didn't achieve the goal of "a shorter and punchier word than 'rectangular solid' for this common concept!"
I suggest "oblong" is the word required.
But I've seen this used to refer to 2-dimensional as well as three-dimensional shapes, as well as to non-rectalinear shapes. Andy
WFL
On 1/30/19, Mike Speciner <ms@alum.mit.edu> wrote:
Actually, it's parallelepiped.
On 30-Jan-19 18:19, Allan Wechsler wrote:
The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped". Some author, I can't remember who, writes "2-box" and "3-box" and the like.
On Wed, Jan 30, 2019 at 6:07 PM Dan Asimov <dasimov@earthlink.net> wrote:
Good point!* I will now try to find if something comparable has been done for more general polyhedra.
—Dan
—————————————————————————————————————————————————————————————— * If you've never tried it, get a fat mass-market paperback (i.e., small format, c. 4"x7"), put a rubber band about it to hold it together, and note the disparate efforts needed in order to get it to spin 360º midair about the three axes perpendicular to the faces of this rectangular solid.**
** "Rectangular solid"??? Surely there is a shorter and punchier word for this common concept!
Andy Latto <andy.latto@pobox.com> écrit: ----- Assuming the die has 3 distinct moments, there are two stable ways to have the die spinning as it is tossed. I don't see any reason to expect the probabilities will be the same for these two different ways to throw the die. Or the unstable ways, but those are harder for the thrower to reproduce accurately. -----
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-- Andy.Latto@pobox.com
Yes, rectangular parellelepiped works too. There are a bunch of other synonyms. You just need to be sure they're sufficiently qualified to restrict the faces to rectangles (and therefore the face angles to 90 degrees, and therefore the dihedral angles to 90 degrees). Regarding fair dice, I only trust solutions where all faces are geometrically identical to each other. For a 6-sided die, you can use a regular dodecahedron, with opposite faces having the same number (or in fact any other numbering with two of each of the numbers from one to six). For a 5-sided die, you can use a regular icosahedron with four of each number. An alternative strategy that works for even-numbers of six or higher is to use a bipyramind. This way you only need one face per number. Unfortunately, if the individual pyramids have odd numbers of faces (i.e., if the total number of faces is 2 mod 4), then you need to either read the result from the face on the bottom, or else paint the result on the exposed upper-edge. Or, you can double the number of faces and duplicate the numbers to avoid this problem. For odd numbers, you could of course use twice the number of faces with duplicate numbers. You could also use a pyramid with n sides, then use a rounded base that forces it to land on a side. You'd have to be very careful to make the base perfectly symmetrical to avoid biasing the direction in which it tips. I don't particularly like this solution, but you could always paint the values on the rounded part, avoiding the problem of having to read them from the edges in the odd case. A more elegant solution is to use curved sides, like the exposed surfaces of a peeled orange. Taper the ends and make them pointy (like an American football) so that it will always rest in the center of one of the sides. This will work for any number that's three or higher, although again, for an odd number of faces, you'd need to either read the answer from an edge or else double the number of faces. Tom Andy Latto writes:
On Wed, Jan 30, 2019 at 8:00 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped" >> I don't think so. See https://en.wikipedia.org/wiki/Parallelepiped
Which is presumably why I was taught "rectangular parallelepiped" as a child, which was fun to say, but certainly didn't achieve the goal of "a shorter and punchier word than 'rectangular solid' for this common concept!"
I suggest "oblong" is the word required.
But I've seen this used to refer to 2-dimensional as well as three-dimensional shapes, as well as to non-rectalinear shapes.
Andy
WFL
On 1/30/19, Mike Speciner <ms@alum.mit.edu> wrote:
Actually, it's parallelepiped.
On 30-Jan-19 18:19, Allan Wechsler wrote:
The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped". Some author, I can't remember who, writes "2-box" and "3-box" and the like.
On Wed, Jan 30, 2019 at 6:07 PM Dan Asimov <dasimov@earthlink.net> wrote:
Good point!* I will now try to find if something comparable has been done for more general polyhedra.
—Dan
—————————————————————————————————————————————————————————————— * If you've never tried it, get a fat mass-market paperback (i.e., small format, c. 4"x7"), put a rubber band about it to hold it together, and note the disparate efforts needed in order to get it to spin 360º midair about the three axes perpendicular to the faces of this rectangular solid.**
** "Rectangular solid"??? Surely there is a shorter and punchier word for this common concept!
Andy Latto <andy.latto@pobox.com> écrit: ----- Assuming the die has 3 distinct moments, there are two stable ways to have the die spinning as it is tossed. I don't see any reason to expect the probabilities will be the same for these two different ways to throw the die. Or the unstable ways, but those are harder for the thrower to reproduce accurately. -----
Actually, using a trapezohedron in place of a bipyramid will eliminate the need to read the answer from an edge in the 2 mod 4 case. Apparently this solution is already in use. https://en.wikipedia.org/wiki/Dice#Variants A similar problem exists for tetrahedral dice, which is solved by writing the number at the base of each exposed face (so each number appears in three places). See also: https://en.wikipedia.org/wiki/Dice#Rarer_variations Tom Tom Karzes writes:
Yes, rectangular parellelepiped works too. There are a bunch of other synonyms. You just need to be sure they're sufficiently qualified to restrict the faces to rectangles (and therefore the face angles to 90 degrees, and therefore the dihedral angles to 90 degrees).
Regarding fair dice, I only trust solutions where all faces are geometrically identical to each other. For a 6-sided die, you can use a regular dodecahedron, with opposite faces having the same number (or in fact any other numbering with two of each of the numbers from one to six). For a 5-sided die, you can use a regular icosahedron with four of each number.
An alternative strategy that works for even-numbers of six or higher is to use a bipyramind. This way you only need one face per number. Unfortunately, if the individual pyramids have odd numbers of faces (i.e., if the total number of faces is 2 mod 4), then you need to either read the result from the face on the bottom, or else paint the result on the exposed upper-edge. Or, you can double the number of faces and duplicate the numbers to avoid this problem.
For odd numbers, you could of course use twice the number of faces with duplicate numbers. You could also use a pyramid with n sides, then use a rounded base that forces it to land on a side. You'd have to be very careful to make the base perfectly symmetrical to avoid biasing the direction in which it tips. I don't particularly like this solution, but you could always paint the values on the rounded part, avoiding the problem of having to read them from the edges in the odd case.
A more elegant solution is to use curved sides, like the exposed surfaces of a peeled orange. Taper the ends and make them pointy (like an American football) so that it will always rest in the center of one of the sides. This will work for any number that's three or higher, although again, for an odd number of faces, you'd need to either read the answer from an edge or else double the number of faces.
Tom
In the UK this shape (product of three intervals) is usually called a 'cuboid' (and is taught as such in primary schools). Consequently, when a (foreign, possibly Eastern European) professor used the term 'rectilinear parallelepiped' to refer to the shape in a Cambridge undergraduate lecture, it appeared somewhat circumlocutory. Ever since, I've had friends humorously refer to the small cuboids of flapjack and millionaire's shortbread as 'parallelepipeds', in the sense of: A: "I'm feeling hungry; shall we get snacks from Sainsbury's?" B: "Sure, what were you thinking?" A: "Parallelepipeds, maybe?" B: "Great!" On a more serious note, I've often seen 'brick' used for the n-dimensional generalisation (an arbitrary finite product of intervals). Best wishes, Adam P. Goucher
Sent: Thursday, January 31, 2019 at 1:44 AM From: "Andy Latto" <andy.latto@pobox.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] What is the probability that a randomly thrown polyhedral die will land on a given face?
On Wed, Jan 30, 2019 at 8:00 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped" >> I don't think so. See https://en.wikipedia.org/wiki/Parallelepiped
Which is presumably why I was taught "rectangular parallelepiped" as a child, which was fun to say, but certainly didn't achieve the goal of "a shorter and punchier word than 'rectangular solid' for this common concept!"
I suggest "oblong" is the word required.
But I've seen this used to refer to 2-dimensional as well as three-dimensional shapes, as well as to non-rectalinear shapes.
Andy
WFL
On 1/30/19, Mike Speciner <ms@alum.mit.edu> wrote:
Actually, it's parallelepiped.
On 30-Jan-19 18:19, Allan Wechsler wrote:
The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped". Some author, I can't remember who, writes "2-box" and "3-box" and the like.
On Wed, Jan 30, 2019 at 6:07 PM Dan Asimov <dasimov@earthlink.net> wrote:
Good point!* I will now try to find if something comparable has been done for more general polyhedra.
—Dan
—————————————————————————————————————————————————————————————— * If you've never tried it, get a fat mass-market paperback (i.e., small format, c. 4"x7"), put a rubber band about it to hold it together, and note the disparate efforts needed in order to get it to spin 360º midair about the three axes perpendicular to the faces of this rectangular solid.**
** "Rectangular solid"??? Surely there is a shorter and punchier word for this common concept!
Andy Latto <andy.latto@pobox.com> écrit: ----- Assuming the die has 3 distinct moments, there are two stable ways to have the die spinning as it is tossed. I don't see any reason to expect the probabilities will be the same for these two different ways to throw the die. Or the unstable ways, but those are harder for the thrower to reproduce accurately. -----
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-- Andy.Latto@pobox.com
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Actually a parallelepiped is a 3d analogue of a parallelogram. The 3d analogue of a rectangle is a right rectangular prism, also known as a rectangular cuboid. Tom Allan Wechsler writes:
The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped". Some author, I can't remember who, writes "2-box" and "3-box" and the like.
On Wed, Jan 30, 2019 at 6:07 PM Dan Asimov <dasimov@earthlink.net> wrote:
Good point!* I will now try to find if something comparable has been done for more general polyhedra.
—Dan
—————————————————————————————————————————————————————————————— * If you've never tried it, get a fat mass-market paperback (i.e., small format, c. 4"x7"), put a rubber band about it to hold it together, and note the disparate efforts needed in order to get it to spin 360º midair about the three axes perpendicular to the faces of this rectangular solid.**
** "Rectangular solid"??? Surely there is a shorter and punchier word for this common concept!
Andy Latto <andy.latto@pobox.com> écrit: ----- Assuming the die has 3 distinct moments, there are two stable ways to have the die spinning as it is tossed. I don't see any reason to expect the probabilities will be the same for these two different ways to throw the die. Or the unstable ways, but those are harder for the thrower to reproduce accurately. -----
OK, my dad was wrong. I was about ten. We had this exact conversation, and I said, "But, Dad, the word sounds like it means something like a parallelogram," and he said, "Yeah, I know, but the accepted definition is that it has rectangular faces." He was a professor of mathematics and I bowed to his authority. If anyone wants to argue with him about it, I can give you the address of Gan Zikaron Cemetery. On Wed, Jan 30, 2019 at 8:15 PM Tom Karzes <karzes@sonic.net> wrote:
Actually a parallelepiped is a 3d analogue of a parallelogram. The 3d analogue of a rectangle is a right rectangular prism, also known as a rectangular cuboid.
Tom
Allan Wechsler writes:
The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped". Some author, I can't remember who, writes "2-box" and "3-box" and the like.
On Wed, Jan 30, 2019 at 6:07 PM Dan Asimov <dasimov@earthlink.net> wrote:
Good point!* I will now try to find if something comparable has been done for more general polyhedra.
—Dan
—————————————————————————————————————————————————————————————— * If you've never tried it, get a fat mass-market paperback (i.e., small format, c. 4"x7"), put a rubber band about it to hold it together, and note the disparate efforts needed in order to get it to spin 360º midair about the three axes perpendicular to the faces of this rectangular solid.**
** "Rectangular solid"??? Surely there is a shorter and punchier word for this common concept!
Andy Latto <andy.latto@pobox.com> écrit: ----- Assuming the die has 3 distinct moments, there are two stable ways to have the die spinning as it is tossed. I don't see any reason to expect the probabilities will be the same for these two different ways to throw the die. Or the unstable ways, but those are harder for the thrower to reproduce accurately. -----
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Tom Karzes <karzes@sonic.net> wrote:
The 3d analogue of a rectangle is a right rectangular prism, also known as a rectangular cuboid.
Also also known as a "brick". --Michael
Allan Wechsler writes:
The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped". Some author, I can't remember who, writes "2-box" and "3-box" and the like.
On Wed, Jan 30, 2019 at 6:07 PM Dan Asimov <dasimov@earthlink.net> wrote:
Good point!* I will now try to find if something comparable has been done for more general polyhedra.
—Dan
—————————————————————————————————————————————————————————————— * If you've never tried it, get a fat mass-market paperback (i.e., small format, c. 4"x7"), put a rubber band about it to hold it together, and note the disparate efforts needed in order to get it to spin 360º midair about the three axes perpendicular to the faces of this rectangular solid.**
** "Rectangular solid"??? Surely there is a shorter and punchier word for this common concept!
Andy Latto <andy.latto@pobox.com> écrit: ----- Assuming the die has 3 distinct moments, there are two stable ways to have the die spinning as it is tossed. I don't see any reason to expect the probabilities will be the same for these two different ways to throw the die. Or the unstable ways, but those are harder for the thrower to reproduce accurately. -----
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-- Forewarned is worth an octopus in the bush.
I like "block" or "3-block". Brent On 1/30/2019 5:13 PM, Tom Karzes wrote:
Actually a parallelepiped is a 3d analogue of a parallelogram. The 3d analogue of a rectangle is a right rectangular prism, also known as a rectangular cuboid.
Tom
Allan Wechsler writes:
The word my father taught me for the three-dimensional analogue of a rectangle is "parallelopiped". Some author, I can't remember who, writes "2-box" and "3-box" and the like.
On Wed, Jan 30, 2019 at 6:07 PM Dan Asimov <dasimov@earthlink.net> wrote:
Good point!* I will now try to find if something comparable has been done for more general polyhedra.
—Dan
—————————————————————————————————————————————————————————————— * If you've never tried it, get a fat mass-market paperback (i.e., small format, c. 4"x7"), put a rubber band about it to hold it together, and note the disparate efforts needed in order to get it to spin 360º midair about the three axes perpendicular to the faces of this rectangular solid.**
** "Rectangular solid"??? Surely there is a shorter and punchier word for this common concept!
Andy Latto <andy.latto@pobox.com> écrit: ----- Assuming the die has 3 distinct moments, there are two stable ways to have the die spinning as it is tossed. I don't see any reason to expect the probabilities will be the same for these two different ways to throw the die. Or the unstable ways, but those are harder for the thrower to reproduce accurately. -----
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participants (9)
-
Adam P. Goucher -
Allan Wechsler -
Andy Latto -
Brent Meeker -
Dan Asimov -
Fred Lunnon -
Michael Kleber -
Mike Speciner -
Tom Karzes