A quickie. No diagram, but no need. Given a unit cube, vertices labeled ABCD on the top face, EFGH on the bottom; where A <--> E, B <--> F, etc. Draw diagonals GD and BD, across the appropriate faces. What is the angle GDB? You have two minutes, no calculator, pen and paper only - -- Rich

# daily puzzle

Started by ●September 14, 2020

Reply by ●September 14, 20202020-09-14

On Monday, September 14, 2020 at 2:49:51 PM UTC-7, RichD wrote:> A quickie. No diagram, but no need. > > Given a unit cube, vertices labeled ABCD on the top face, > EFGH on the bottom; where A <--> E, B <--> F, etc. > Draw diagonals GD and BD, across the appropriate faces. > > What is the angle GDB?Maybe closer to the subject of this group, what is the symmetry of the 111 face of a silicon crystal. (Or any cubic crystal.) With the right coordinate system, the coordinates of the points in question are [110], [101], and [011]. However, the answer is even easier, as there is only one angle that is a possible solution, and that we should know without the need to look it up.

Reply by ●September 15, 20202020-09-15

On Tuesday, September 15, 2020 at 6:49:51 AM UTC+9, RichD wrote:> A quickie. No diagram, but no need. > > Given a unit cube, vertices labeled ABCD on the top face, > EFGH on the bottom; where A <--> E, B <--> F, etc. > Draw diagonals GD and BD, across the appropriate faces. > > What is the angle GDB? > > You have two minutes, no calculator, pen and paper only - > > -- > Rich60 deg?

Reply by ●October 10, 20202020-10-10

On 15.09.2020 0:49, RichD wrote:> A quickie. No diagram, but no need. > > Given a unit cube, vertices labeled ABCD on the top face, > EFGH on the bottom; where A <--> E, B <--> F, etc. > Draw diagonals GD and BD, across the appropriate faces. > > What is the angle GDB? > > You have two minutes, no calculator, pen and paper only - > > -- > Rich >I'm late to this, but points G, D and B form an equilateral triangle while (obviously) lying on the same plane, so the answer is 120 degrees. -- Gene

Reply by ●October 10, 20202020-10-10

On 10.10.2020 14:50, Gene Filatov wrote:> On 15.09.2020 0:49, RichD wrote: >> A quickie. No diagram, but no need. >> >> Given a unit cube, vertices labeled ABCD on the top face, >> EFGH on the bottom; where A <--> E, B <--> F, etc. >> Draw diagonals GD and BD, across the appropriate faces. >> >> What is the angle GDB? >> >> You have two minutes, no calculator, pen and paper only - >> >> -- >> Rich >> > > I'm late to this, but points G, D and B form an equilateral triangle > while (obviously) lying on the same plane, so the answer is 120 degrees. > > -- > GeneCan't believe I'm that stupid... :) 60 degrees. -- Gene

Reply by ●October 10, 20202020-10-10

On Saturday, October 10, 2020 at 4:52:43 AM UTC-7, Gene Filatov wrote: (snip)> Can't believe I'm that stupid... :) 60 degrees.> GeneLooking along a cube diagonal, a cube has hexagonal symmetry. If you pack spheres into a single layer, they form in equilateral triangles, and the layer has hexagonal symmetry. If you stack such layers, such that spheres on each layer go into the spaces between spheres on the layer below, there are three ways to stack each over the previous layer. If you name those three A, B, C, then the stacking patterns can be named. If you stack A, B, C, A, B, C, etc. the result has hexagonal symmetry, and the crystal form is named HCP for hexagonal close packing. In this case, the layer spacing can be, but isn't required to be, such that the spacing between adjacent atoms on different layers is equal to that within a layer. If you stack A, B, A, B, etc., the result is face-centered cubic. This is the symmetry of looking at a cube along its diagonal. Cubic crystals have the same index of refraction in any direction. Others don't, and are then birefringent, which causes different light polarizations to have different index of refraction. Cubic zirconia is used for jewelry instead of the alternative, hexagonal form, as it isn't birefringent. Diamond has cubic symmetry, but it is possible to put carbon atoms together in a diamond-like structure with hexagonal symmetry.

Reply by ●October 10, 20202020-10-10

On 10.10.2020 17:13, ga...@u.washington.edu wrote:> On Saturday, October 10, 2020 at 4:52:43 AM UTC-7, Gene Filatov wrote: > > (snip) > >> Can't believe I'm that stupid... :) 60 degrees. > >> Gene > > Looking along a cube diagonal, a cube has hexagonal symmetry. > > If you pack spheres into a single layer, they form in equilateral triangles, > and the layer has hexagonal symmetry. > > If you stack such layers, such that spheres on each layer go into the > spaces between spheres on the layer below, there are three ways to > stack each over the previous layer. If you name those three A, B, C, > then the stacking patterns can be named. > > If you stack A, B, C, A, B, C, etc. the result has hexagonal symmetry, > and the crystal form is named HCP for hexagonal close packing. > In this case, the layer spacing can be, but isn't required to be, > such that the spacing between adjacent atoms on different layers > is equal to that within a layer. > > If you stack A, B, A, B, etc., the result is face-centered cubic. > This is the symmetry of looking at a cube along its diagonal. > > Cubic crystals have the same index of refraction in any direction. > Others don't, and are then birefringent, which causes different > light polarizations to have different index of refraction. > > Cubic zirconia is used for jewelry instead of the alternative, > hexagonal form, as it isn't birefringent. Diamond has > cubic symmetry, but it is possible to put carbon atoms > together in a diamond-like structure with hexagonal > symmetry. >It's a special kind of a skill to explain complex issues using simple terms! I love how one book* on defects in crystals dealt with vacancies. As you know, a vacancy is a point defect, in which an atom is missing from its place in the lattice. Vacancies increase the energy of the crystal, but decrease the entropy, so there's always a certain amount of vacancies that minimizes the free energy; that amount is known to increase with temperature. When you raise the temperature, new vacancies are formed at the surface and move to the inner layers of the crystal. There, I was struck by a particular wording: "the crystal sort of dissolves the void". It's not even physics, it's poetry! ;-) [*] Novikov I.I., Defects of crystalline structure of metals (Metallurgiya, Moscow, 1975). -- Gene

Reply by ●October 10, 20202020-10-10

On 11.10.2020 3:02, Gene Filatov wrote:> On 10.10.2020 17:13, ga...@u.washington.edu wrote: >> On Saturday, October 10, 2020 at 4:52:43 AM UTC-7, Gene Filatov wrote: >> >> (snip) >> >>> Can't believe I'm that stupid... :) 60 degrees. >> >>> Gene >> >> Looking along a cube diagonal, a cube has hexagonal symmetry. >> >> If you pack spheres into a single layer, they form in equilateral >> triangles, >> and the layer has hexagonal symmetry. >> >> If you stack such layers, such that spheres on each layer go into the >> spaces between spheres on the layer below, there are three ways to >> stack each over the previous layer. If you name those three A, B, C, >> then the stacking patterns can be named. >> >> If you stack A, B, C, A, B, C, etc. the result has hexagonal symmetry, >> and the crystal form is named HCP for hexagonal close packing. >> In this case, the layer spacing can be, but isn't required to be, >> such that the spacing between adjacent atoms on different layers >> is equal to that within a layer. >> >> If you stack A, B, A, B, etc., the result is face-centered cubic. >> This is the symmetry of looking at a cube along its diagonal. >> >> Cubic crystals have the same index of refraction in any direction. >> Others don't, and are then birefringent, which causes different >> light polarizations to have different index of refraction. >> >> Cubic zirconia is used for jewelry instead of the alternative, >> hexagonal form, as it isn't birefringent. Diamond has >> cubic symmetry, but it is possible to put carbon atoms >> together in a diamond-like structure with hexagonal >> symmetry. >> > > It's a special kind of a skill to explain complex issues using simple > terms! > > I love how one book* on defects in crystals dealt with vacancies. As you > know, a vacancy is a point defect, in which an atom is missing from its > place in the lattice. Vacancies increase the energy of the crystal, but > decrease the entropy, so there's always a certain amount of vacancies > that minimizes the free energy; that amount is known to increase with > temperature. When you raise the temperature, new vacancies are formed at > the surface and move to the inner layers of the crystal. There, I was > struck by a particular wording: "the crystal sort of dissolves the > void". It's not even physics, it's poetry! ;-) > > [*] Novikov I.I., Defects of crystalline structure of metals > (Metallurgiya, Moscow, 1975). > > -- > Gene >Oops... of course, vacancies _increase_ entropy, but as the free energy is F = E - TS, they decrease the latter term. It's actually hard to explain anything. Definitely requires a special skill. ;-) -- Gene.