daily puzzle

[RichD]

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

[[e-mail]]

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.

[pulpo]

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

60 deg?

[Gene Filatov]

Post by RichD
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

[Gene Filatov]

Post by Gene Filatov

Post by RichD
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

Can't believe I'm that stupid... :) 60 degrees.

--
Gene

[[e-mail]]

On Saturday, October 10, 2020 at 4:52:43 AM UTC-7, Gene Filatov wrote:

(snip)

Post by Gene Filatov
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.

[Gene Filatov]

Post by ***@u.washington.edu
(snip)

Post by Gene Filatov
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

[Gene Filatov]

Post by Gene Filatov

Post by ***@u.washington.edu
(snip)

Post by Gene Filatov
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.