Static universe - reply

On Apr 8, 6:49*pm, Eric Gisse wrote:
On Apr 7, 11:52 pm, davd wrote:
[...]

1) Your 'analysis' of the Tolman surface brightness test from Lubin &
Sandage amounts to reading their conclusion and saying 'nuh-uh!' You
even *admit* that if the evolutionary relationship cited in the paper
was used, that the agreement was excellent. Which in the very next
sentence, without justification, you toss out the window to argue that
there is no agreement.

I am only taking the trouble to reply to you because the moderator
said that "there is quite a lot of well-reasoned content".
This is a complete misrepresentation of what I said. Let me repeat.

Doesn't seem all that complete.

Except for a very small disagreement about some numerical values which
is irrelevant to the final analysis I completely agree with Lubin &
Sandage's analysis including the *required luminosity evolution. What
I disagree with them is that this evolution is reasonable.

Yes, you disagree. You do not give a credible reason as to why the
evolutionary models are incorrect, you just toss them out and use the
raw value then proclaim victory.

The disagreement is made slightly less credible by you not
understanding that the (1+z)^-4 factor in the apparent luminosity
comes from. It is two factors of (1+z)^-2, each from time dilation and
expansion. Misrepresenting and misunderstanding a theory then basing
further claims off that is a very dubious and dishonest proposition.
It is funny that I state this in the paper!

Furthermore
is the fact that the nearby galaxies are brightest cluster galaxies
(BCG) whereas the distant galaxies are normal cluster galaxies. It is
well known that BCG galaxies are significantly larger and brighter
than the expected largest member of a cluster. They did not discuss
this so that I presume that they thought that the Kormendy correction
(the strong dependence of luminosity on absolute galactic radius)
would take care of the BCG mismatch. Using the BB angular size
function the distant galaxies are on average approximately 0.26 time
the size of the nearby galaxies which is much smaller than that
expected from the BCG discrepancy. The ratio is 0.36 if a (1+z) factor
is removed from the BB angular size function. With this correction I
showed that the observed exponent for the surface brightness as a
function of redshift is 1.03+\-0.16 which is in excellent agreement
withe the expected value of one.

And your justification for arbitrarily tinkering with the functions is
what, exactly?
Arbitrary tinkering? It is clearly explained

Keep in mind your 'expected value' does not agree with the raw value
either, so you'll need a nice large correction for your theory.
It does agree within the paradigm of a static universe.

Lubin & Sandage did examine a
particular model for a tired light cosmology and claimed that it was
inconsistent. My approach is much simpler and shows that a tired light
cosmology can be completely consistent with these surface brightness
observations. Finally the observed exponent using BB is 2.16+\-0.13 to
be compared with an expected value of 4. This requires a luminosity
evolution exponent of 1.84 which is very large. Without strong outside
evidence this seems unlikely.

Your entire argument rests on the wishing and hoping that there are
zero evolutionary differences for multiple different galactic
clusters. You have not justified your claim that it is 'very large',
at least in the sense that you think it is larger than it ought to
be.
What do you mean by "zero evolutionary differences for multiple
different galactic clusters" What about galaxy interactions. It seems
that a significant number of galaxies have had collisions that may
have reset their evolutionary clock.

There's a difference of /_\z ~ 0.1 between each cluster. Short of
looking it up myself to confirm, I can tell you that those clusters
are not gravitationally bound and are rather distant to the tune of
~billion light years. That would be a radial distance, at least.
Exactly how distant would require a literature search, but I am
convinced that they are far enough away that there's no sane reason to
argue that they are all at the exact same point in their evolutionary
history which is EXACTLY what you are trying to argue whether you know
it or not.
???




2) Your 'analysis' of the various CMBR temperature measurements at non-
local distances is literally nothing but you saying 'but physics is
hard! I don't believe you!'

What utter rubbish! You have failed to understand the argument that I
made.

What argument? Not a rhetorical question. There is no reasoned
argument for me to 'not understand'.

You point out that the column densities of the plasma need to be
reasonably well known, then you wander off and say 'well in MY THEORY,
light behaves in a fashion NEVER BEFORE SEEN in a plasma so I am going
to be skeptical!

Not a verbatim quote, but close enough. You use your theory to inject
false uncertainty in multiple independent measurements at various
redshifts, using arguments that do not have any basis in current
electromagnetic theory.

If we are arguing about which cosmological theory best fits the
observations we must be consistent about keeping each argument within
its own paradigm. It is this self consistency that is important.


Seriously. This is your argument. No discussion of possible systematic
errors, or how sensitive the observation is to various assumed models
or even if that could be a factor. Just 'this is hard' and then you
move on.

3) Your entire SN1a discussion is rather odd.

I remember seeing some figures regarding Kowalski, et. al., ApJ (2008)
before in a previous unrelated discussion. I could have SWORN that
there were a rather solid amount of data points extending up to z=1.6.
The project website and the relevant data slide (http://supernova.lbl.gov/Union/figur...agramSlide.pdf
) makes me wonder what's going on as the Union1 data set goes up to
z=1.6.

Clearly you have failed to properly read the axis label it is
2.5log(1+z) and not z.

I noticed that. Not relevant.

What is relevant is immediately previous was your cutoff of data
points at z = 1.139 in the binning discussion. You repeat the explicit
Only for that particular argument about expected densities.
nature of the cutoff again in Table 6.

Now if you'd like to argue you did, in fact, use the whole data set
you'll seriously need to rewrite that section. Like, for example, why
you need to bin the data and cut it off at arbitrary redshifts and
magnitudes when the point of the exercise is to test the validity of a
certain power law argument whose relevance is somewhat cryptic.

Especially when you write this: "The results for bin six show that 33
out of 50 supernovae had an apparent magnitude brighter than the cut-o
ff".

What IS your point in that section, anyway? You don't even show the
binning, or explain why the bin sizes are meaningful, or justify the
procedure at all. You pick an arbitrary cutoff, and some members of a
particular bin go past it. I'm just not seeing the significance.

If you read the paper the binning is described in the description for
table 6. Later usage refers directly to this description.

If I have to go through the dataset myself to figure out what the hell
you are talking about, you need to revisit your presentation of the
material.



You then argue that, in a bizarre and nonsensical fashion, that the
lack of high-z supernovae is a problem for the concordance cosmology.
Which struck me as odd not just because that's wrong, but because in
the previous breath you were pointing out that there haven't been a
large amount of searches for high-z supernovae.
I never said that!

Via Section 4.3.1:

Exhibit A: "The results for bin one are not unexpected. It simply
shows that there have been many more searches done for supernovae at
nearby redshifts."

[You never actually show the binning]
See above.

Exhibit B: "The problem with the BB results is that there is a
dramatic shortage of supernovae in the high redshift bins."

Now you probably could argue that you never actually SAID that there
have not been a lot of high-z supernovae searches but I thought the
implication was somewhat reasonable.

You are arguing out of both sides of your mouth, to hedge your bets.
You know damn well that quality light curves from high-z type 1a
events are hard to find.
If supernovae are well above the apparent magnitude cutoff why should
observations of the light curve be difficult.


[...]

I note that you fail to criticize the analysis for gamma
ray bursts, galaxy luminosity functions and quasar luminosity
functions.

I'm grabbing the low hanging fruit. I don't see the point of engaging
an argument about a facet of the subject I don't understand too well
when you have multitudes of problems in the areas I do know reasonably
well.

How convenient. Since the analyses for these objects are independent
of each other a significant disagreement of any one of them with
expansion is a serious problem for the current cosmological model.


You don't even have an analysis in the cases I've looked at - you just
take published papers and say 'nuh-uh'.

On Apr 13, 5:29 am, davd wrote:
On Apr 8, 6:49 pm, Eric Gisse wrote:







On Apr 7, 11:52 pm, davd wrote:
[...]

1) Your 'analysis' of the Tolman surface brightness test from Lubin &
Sandage amounts to reading their conclusion and saying 'nuh-uh!' You
even *admit* that if the evolutionary relationship cited in the paper
was used, that the agreement was excellent. Which in the very next
sentence, without justification, you toss out the window to argue that
there is no agreement.

I am only taking the trouble to reply to you because the moderator
said that "there is quite a lot of well-reasoned content".
This is a complete misrepresentation of what I said. Let me repeat.

Doesn't seem all that complete.

Except for a very small disagreement about some numerical values which
is irrelevant to the final analysis I completely agree with Lubin &
Sandage's analysis including the required luminosity evolution. What
I disagree with them is that this evolution is reasonable.

Yes, you disagree. You do not give a credible reason as to why the
evolutionary models are incorrect, you just toss them out and use the
raw value then proclaim victory.

The disagreement is made slightly less credible by you not
understanding that the (1+z)^-4 factor in the apparent luminosity
comes from. It is two factors of (1+z)^-2, each from time dilation and
expansion. Misrepresenting and misunderstanding a theory then basing
further claims off that is a very dubious and dishonest proposition.

It is funny that I state this in the paper!

Well, yes, otherwise I wouldn't be commenting on it.

You claim there is one factor of (1+z) each from time dilation and
expansion, and two from aberration. That's just wrong.










Furthermore
is the fact that the nearby galaxies are brightest cluster galaxies
(BCG) whereas the distant galaxies are normal cluster galaxies. It is
well known that BCG galaxies are significantly larger and brighter
than the expected largest member of a cluster. They did not discuss
this so that I presume that they thought that the Kormendy correction
(the strong dependence of luminosity on absolute galactic radius)
would take care of the BCG mismatch. Using the BB angular size
function the distant galaxies are on average approximately 0.26 time
the size of the nearby galaxies which is much smaller than that
expected from the BCG discrepancy. The ratio is 0.36 if a (1+z) factor
is removed from the BB angular size function. With this correction I
showed that the observed exponent for the surface brightness as a
function of redshift is 1.03+\-0.16 which is in excellent agreement
withe the expected value of one.

And your justification for arbitrarily tinkering with the functions is
what, exactly?

Arbitrary tinkering? It is clearly explained

Yes, it is clearly explained that the number you picked gives you the
answer you want. Beyond that, you have no actual justification.


Keep in mind your 'expected value' does not agree with the raw value
either, so you'll need a nice large correction for your theory.

It does agree within the paradigm of a static universe.

For the most literal value of 'static', where nothing ever moves or
evolves through time. Because that's the only way it makes sense,
given the redshift difference of 0.1 between each of the Tolman test
galaxies.


Lubin & Sandage did examine a
particular model for a tired light cosmology and claimed that it was
inconsistent. My approach is much simpler and shows that a tired light
cosmology can be completely consistent with these surface brightness
observations. Finally the observed exponent using BB is 2.16+\-0.13 to
be compared with an expected value of 4. This requires a luminosity
evolution exponent of 1.84 which is very large. Without strong outside
evidence this seems unlikely.

Your entire argument rests on the wishing and hoping that there are
zero evolutionary differences for multiple different galactic
clusters. You have not justified your claim that it is 'very large',
at least in the sense that you think it is larger than it ought to
be.

What do you mean by "zero evolutionary differences for multiple
different galactic clusters" What about galaxy interactions. It seems
that a significant number of galaxies have had collisions that may
have reset their evolutionary clock.

To the same value? Across multiple galaxies?

Don't think so.


There's a difference of /_\z ~ 0.1 between each cluster. Short of
looking it up myself to confirm, I can tell you that those clusters
are not gravitationally bound and are rather distant to the tune of
~billion light years. That would be a radial distance, at least.
Exactly how distant would require a literature search, but I am
convinced that they are far enough away that there's no sane reason to
argue that they are all at the exact same point in their evolutionary
history which is EXACTLY what you are trying to argue whether you know
it or not.

???

Fine, small words: Galaxies far apart live different lives.












2) Your 'analysis' of the various CMBR temperature measurements at non-
local distances is literally nothing but you saying 'but physics is
hard! I don't believe you!'

What utter rubbish! You have failed to understand the argument that I
made.

What argument? Not a rhetorical question. There is no reasoned
argument for me to 'not understand'.

You point out that the column densities of the plasma need to be
reasonably well known, then you wander off and say 'well in MY THEORY,
light behaves in a fashion NEVER BEFORE SEEN in a plasma so I am going
to be skeptical!

Not a verbatim quote, but close enough. You use your theory to inject
false uncertainty in multiple independent measurements at various
redshifts, using arguments that do not have any basis in current
electromagnetic theory.

If we are arguing about which cosmological theory best fits the
observations we must be consistent about keeping each argument within
its own paradigm. It is this self consistency that is important.

Yeah, well, you don't get to just go up and say a bunch of independent
measurements from multiple authors are not credible 'because my theory
doesn't agree with them'. No, saying 'but the measurement is hard!'
doesn't discredit the result either, it just means it is worth
thinking about the assumptions used.

Typically it is observation that discredits theory, but perhaps I was
taught the scientific theory incorrectly.

You have no credible reason to disregard the observations, but you do
so anyway purely because it proves your theory false. Now remind me,
what's the difference between a scientist and a crank?

[...]

Exhibit B: "The problem with the BB results is that there is a
dramatic shortage of supernovae in the high redshift bins."

Now you probably could argue that you never actually SAID that there
have not been a lot of high-z supernovae searches but I thought the
implication was somewhat reasonable.

You are arguing out of both sides of your mouth, to hedge your bets.
You know damn well that quality light curves from high-z type 1a
events are hard to find.

If supernovae are well above the apparent magnitude cutoff why should
observations of the light curve be difficult.

Quick primer on Type 1a supernovae. The reason they are ever-so-handy
is that, within a relatively tight margin, they have a constant
absolute luminosity. Now, the amazing thing about an expanding
universe is that if you have a little SN1a blinker out there at
increasingly further distances it is going to be increasingly harder
to SEE it by virtue of the factor of (1+z)^-4 that drags the apparent
luminosity down into the toilet.

It is a confirmation of the big bang theory that there are less SN1a's
as you go further out, contrary to your goofy belief otherwise, given
all other things being equal that they are VERY HARD TO SEE due to a
reason I just mentioned.




[...]

I note that you fail to criticize the analysis for gamma
ray bursts, galaxy luminosity functions and quasar luminosity
functions.

I'm grabbing the low hanging fruit. I don't see the point of engaging
an argument about a facet of the subject I don't understand too well
when you have multitudes of problems in the areas I do know reasonably
well.

How convenient. Since the analyses for these objects are independent
of each other a significant disagreement of any one of them with
expansion is a serious problem for the current cosmological model.

Except there isn't disagreement. The only one you can say is even
tenuous would be the Tolman surface brightness test, and that's only
because of the modeling difficulty of taking into account the
evolution of the galaxies.










You don't even have an analysis in the cases I've looked at - you just
take published papers and say 'nuh-uh'.

In article , Eric Gisse
writes:

Quick primer on Type 1a supernovae. The reason they are ever-so-handy
is that, within a relatively tight margin, they have a constant
absolute luminosity. Now, the amazing thing about an expanding
universe is that if you have a little SN1a blinker out there at
increasingly further distances it is going to be increasingly harder
to SEE it by virtue of the factor of (1+z)^-4 that drags the apparent
luminosity down into the toilet.

(1+z)^-4 refers to bolometric surface brightness and is independent of
the cosmological model. However, SURFACE brightness isn't an issue in
identifying supernovae (though it is if one needs to identify the host
galaxy). Supernovae do get fainter at higher redshift, but not as
quickly as (1+z)^-4, and the details depend on the cosmological
parameters (otherwise we couldn't use them to measure the cosmological
parameters).

On Apr 14, 8:31 am, Eric Gisse wrote:
You claim there is one factor of (1+z) each from time dilation and
expansion, and two from aberration. That's just wrong.
Since i made a direct quote from Peebles in "Principles of Physical
Cosmology", you must believe that he and other authors are also
wrong!

Yes, it is clearly explained that the number you picked gives you the
answer you want. Beyond that, you have no actual justification.
Where and what?

To the same value? Across multiple galaxies?

Don't think so.
Of course the why would evolution be a function of z.

Fine, small words: Galaxies far apart live different lives.

But you require them to have the same dependence of evolution on
redshift.

Yeah, well, you don't get to just go up and say a bunch of independent
measurements from multiple authors are not credible 'because my theory
doesn't agree with them'. No, saying 'but the measurement is hard!'
doesn't discredit the result either, it just means it is worth
thinking about the assumptions used.

Typically it is observation that discredits theory, but perhaps I was
taught the scientific theory incorrectly.

Precisely I have no argument with the observations but you seem to
think that they are in excellent agreement with an expanding
cosmology. I beg to differ. Show me where my analysis is incorrect.

Quick primer on Type 1a supernovae. The reason they are ever-so-handy
is that, within a relatively tight margin, they have a constant
absolute luminosity. Now, the amazing thing about an expanding
universe is that if you have a little SN1a blinker out there at
increasingly further distances it is going to be increasingly harder
to SEE it by virtue of the factor of (1+z)^-4 that drags the apparent
luminosity down into the toilet.
Please note that I referred to apparent luminosity. For local
supernovae a constant luminosity is the same as constant energy. Where
does your luminosity factor come from? Are you confusing supernovae
luminosity and surface brightness.


It is a confirmation of the big bang theory that there are less SN1a's
as you go further out, contrary to your goofy belief otherwise, given
all other things being equal that they are VERY HARD TO SEE due to a
reason I just mentioned.
If their apparent magnitude is well above the magnitude limit of the
telescope why are they harder to see.

Except there isn't disagreement. The only one you can say is even
tenuous would be the Tolman surface brightness test, and that's only
because of the modeling difficulty of taking into account the
evolution of the galaxies.

But I claim that for these objects there is disagreement and I
provided an independent analysis for each object. For example the
analysis for quasars clearly shows in the figure a very poor agreement
with the standard model.

[Mod. note: quoted text trimmed. Please do not quote text except where
you are directly replying to it -- mjh]

On Apr 14, 2:10 am, davd wrote:
On Apr 14, 8:31 am, Eric Gisse wrote:
You claim there is one factor of (1+z) each from time dilation and
expansion, and two from aberration. That's just wrong.

Since i made a direct quote from Peebles in "Principles of Physical
Cosmology", you must believe that he and other authors are also
wrong!

Aberration implies *proper motion*.

You could make the case that through calculation of luminosity
distance, you pick up one factor of (1+z) through time dilation and
expansion each. Then through the calculation of angular size you pick
up a factor of (1+z)^2 [summarizing a longer argument here], which is
something that is correct. However, aberration does not play here.

Might want to revisit your source and read the quote and its' context
*very* carefully.


Yes, it is clearly explained that the number you picked gives you the
answer you want. Beyond that, you have no actual justification.

Where and what?

To the same value? Across multiple galaxies?

Don't think so.

Of course the why would evolution be a function of z.

Fine, small words: Galaxies far apart live different lives.

But you require them to have the same dependence of evolution on
redshift.

Are all the galaxies in the local group at the same points in their
evolutionary path?

The local group is a gravitationally bound structure, so it is local
for cosmological purposes.


Yeah, well, you don't get to just go up and say a bunch of independent
measurements from multiple authors are not credible 'because my theory
doesn't agree with them'. No, saying 'but the measurement is hard!'
doesn't discredit the result either, it just means it is worth
thinking about the assumptions used.

Typically it is observation that discredits theory, but perhaps I was
taught the scientific theory incorrectly.

Precisely I have no argument with the observations but you seem to
think that they are in excellent agreement with an expanding
cosmology. I beg to differ. Show me where my analysis is incorrect.

The problem is you don't have much in the way of analysis. I'm
repeating myself here, which is getting boring.

You reject the non-local CMB measurements. It has already been
established that there are multiple observations (you have been given
three now) with definitive error bars which with the local data point
of z=0 is more than sufficient to establish the validity of the CMB
being warmer as you look further into the past.

You reject the Tolman surface brightness test. It has been established
that your argument hinges on the rather dubious notion of no galactic
evolution, which the authors kinda seriously disagree with.

You reject the SN1a arguments for no rational reason. They do not
support the static universe model, no matter how you bin and massage
the data.


Quick primer on Type 1a supernovae. The reason they are ever-so-handy
is that, within a relatively tight margin, they have a constant
absolute luminosity. Now, the amazing thing about an expanding
universe is that if you have a little SN1a blinker out there at
increasingly further distances it is going to be increasingly harder
to SEE it by virtue of the factor of (1+z)^-4 that drags the apparent
luminosity down into the toilet.

Please note that I referred to apparent luminosity. For local
supernovae a constant luminosity is the same as constant energy. Where
does your luminosity factor come from? Are you confusing supernovae
luminosity and surface brightness.

Fine, I'll expand the argument further.

An object some cosmological distance away has local luminosity "L".
The relation between the absolute (local) luminosity and the measured
luminosity "F" is L = 4\pi F d_L^2 where d_L is the luminosity
distance to the external observer.

The quantity 4\pi d_L^2 is the surface area of the sphere that
encompasses all the emitted photons, which will be called "A".

For non-local distances, the measured luminosity is going to get
smaller. One factor of (1+z) each from expansion (directly makes A
bigger) and time dilation (which comes from the time dilation of
photon emission directly). This gives us F/L = 1 / A (1+z)^2. You can
calculate the surface area directly from the FRW metric, but that is
more involved than is needed here.

Now, this surface area covers _the entire sphere_. We don't observe
that - we observe a tiny piece of it. That little window laterally
shrinks due to expansion, and if you imagine the window is a small
square in (x,y,z) [with z pointing along the line between you and the
source] it is rather easy to convince yourself that there is another
factor of (1+z) for each side.

Thus the falloff in luminosity goes as proportional to (1+z)^-4.




It is a confirmation of the big bang theory that there are less SN1a's
as you go further out, contrary to your goofy belief otherwise, given
all other things being equal that they are VERY HARD TO SEE due to a
reason I just mentioned.

If their apparent magnitude is well above the magnitude limit of the
telescope why are they harder to see.

Why are you asking this?

An object of absolute luminosity locally and the same object at z = 1
is going to be 1/16 less luminous.

The same object at z = 1.5 is about 39 times less luminous.

At z = 2, 81 times.

Are you still unsure why things that are far away are hard to see?


Except there isn't disagreement. The only one you can say is even
tenuous would be the Tolman surface brightness test, and that's only
because of the modeling difficulty of taking into account the
evolution of the galaxies.

But I claim that for these objects there is disagreement and I
provided an independent analysis for each object. For example the
analysis for quasars clearly shows in the figure a very poor agreement
with the standard model.

I like the bits where you copy and paste from the SDSS data release
papers, without quoting. You would have been better off not calling
attention to this section.

Personally, I find poorly explained graphs rather unconvincing. That,
and your principle assumption is dubious at best. Eg, http://arxiv.org/abs/0704.0806

Sure I know you cited DR3 and that's DR5 but I'm too lazy to care
right this second.

I note two things of worth from the paper: One: The intriguing
similarities of language between the abstract and section 4.6 of your
paper. Two: Figure 6. That looks a lot more like a Poisson
distribution, and most certainly not Gaussian. Doubt the figure has
changed significantly since the last few releases...

You throw out an arbitrary curve for the "distance modulus" in your
Figure 4, except there was absolutely zero discussion of how you
obtained the big bang theory's prediction. No external references were
cited for that, no discussion of the various cosmological parameters.
Nothing.

You do not have an analysis to disagree with. It is difficult to take
you seriously at this point.

[[Mod. note -- 3 excessively-quoted lines snipped here. -- jt]]

On Apr 14, 1:34 am, Phillip Helbig---undress to reply
wrote:
In article , Eric Gisse

writes:
Quick primer on Type 1a supernovae. The reason they are ever-so-handy
is that, within a relatively tight margin, they have a constant
absolute luminosity. Now, the amazing thing about an expanding
universe is that if you have a little SN1a blinker out there at
increasingly further distances it is going to be increasingly harder
to SEE it by virtue of the factor of (1+z)^-4 that drags the apparent
luminosity down into the toilet.

(1+z)^-4 refers to bolometric surface brightness and is independent of
the cosmological model.

That is incorrect. The factor of (1+z)^-4 proportionality between the
absolute luminosity is highly model dependent. Models that don't
include expansion have different proportionalities between apparent
and absolute luminosity.

However, SURFACE brightness isn't an issue in
identifying supernovae (though it is if one needs to identify the host
galaxy). Supernovae do get fainter at higher redshift, but not as
quickly as (1+z)^-4, and the details depend on the cosmological
parameters (otherwise we couldn't use them to measure the cosmological
parameters).

Distances aren't the concern here, its' the luminosity falloff.

In article
,
Eric Gisse writes:

(1+z)^-4 refers to bolometric surface brightness and is independent of
the cosmological model.

That is incorrect. The factor of (1+z)^-4 proportionality between the
absolute luminosity is highly model dependent. Models that don't
include expansion have different proportionalities between apparent
and absolute luminosity.

OK, I should have said "within the framework of cosmology based on the
Robertson-Walker metric, using the Friedmann-Lema?tre equations etc".
No aborigine dream cosmology, no Zeus, no angels pushing crystal
spheres.