Saturday, 8 September 2018

An image to help with Spherical Layers

This relates to the previous post, Spherical Layers.


Each new colour is a new layer.

Of course this is just about layering circles, but the concept of circular layers applies also to spherical layers.  Imagine an incrementally larger circle and how many circles can fit into that circle, or an incrementally larger sphere and how many spheres can fit into that sphere.  As your circle or sphere gets larger, the resultant approximation of the layers gets closer and closer to a circle or a sphere.  In between approximations of circles or spheres, you do get approximations of hexagons or dodecahedrons (which in themselves could be thought of as rough approximations of circles and spheres).

Note that the red and light green layers are also approximations of a dodecagon.  Note also that, when considering polygons, the best approximation of a circle is a regular ∞-gon (or apeirogon), but do note that a circle doesn't have sides per se, it has one curved side (singular, not plural) and is not a polygon.

Thursday, 6 September 2018

Spherical Layers


Say you have a standard solid, rigid sphere like a ball bearing.  Surround that sphere with as many identical spheres as you possibly can.  Hint: the maximum number of equal sized spheres that you can put around a single sphere is 12, according to sphere packing geometry.


You can see how that works here, if you imagine removing the top orange and adding three oranges below, so you have three above, three below and six surrounding the central orange in the middle, for a total of twelve.

Call this the first layer, or Layer 1, and then keep adding more layers.

How many spheres in total will you have when you reach Layer 100?  For bonus points, how many spheres will there be in Layer 100?

(And for extra extra points, is there a formula to calculate the number of spheres with N layers that is more than the just the summation of all spheres in all the layers plus one [for the first sphere]?)

Tuesday, 4 September 2018

A Question of Cosmology


There is such a thing as Hubble time, which is simply the inverse of the Hubble parameter (H).  The inverse of the Hubble constant (H0) is the current Hubble time (because the Hubble constant is the current value of the Hubble parameter).

As I noted back in 2014, in Is the Universe Expanding at the Speed of Light?, the current value of Hubble time is interesting because it’s basically the same as the age of the universe.  Now, because the (current) value of Hubble time is the inverse of the Hubble constant, it varies with measurements of the Hubble constant.

In Is the Universe Expanding at the Speed of Light?, I referred to the values of Hubble constant that were currently available.  These were:
  •  2011 (Hubble) ~71.5 to ~76 km/s/Mpc
  • 2012 (Spitzer) ~72 to ~76.5 km/s/Mpc
  • 2012 (WMAP – after 9 years) 68.52-70.12 km/s/Mpc
  • 2013 (Planck – after four years) 67.03 to 68.57 km/s/Mpc


By happy coincidence, Skydive Phil released a video on “The Hubble Tension” at about the same time as my retrospective podcast listening of the Infinite Monkey Cage got me thinking about the Hubble constant and the age of the universe all over again.  What was bothering me was that there were constant references to the acceleration of the expansion of the universe, together with the assertion (claim, reminder, stating) of the fact that the universe is 13 point something (7 or 8) billion years old (might have been raised in this specific episode or this one).

If the rate at which the universe is expanding is accelerating, then surely the age of the universe comes into question.  The value given for age of the universe has not changed significantly since 2014, when I reported it as 13.8 billion years – the current values range from 13.772 (WMAP) to 13.813 (Planck 2015 data) km/s/Mpc.  The value of the Hubble constant on the other hand …

  • 2018 (Planck) 67.66 (67.24 to 68.08) km/s/Mpc
  • 2018 (Hubble and Gaia) 73.52 (71.90 to 75.14) km/s/Mpc
  • 2017 (LIGO and Virgo) 70.0 (62.0 to 82.0) km/s/Mpc
  • 2016 (eBOSS – after two years) 67.6 (67.0 to 68.3) km/s/Mpc


Skydive Phil’s video focusses on the difference between the eBOSS (baryonic acoustic oscillation) and Planck measurements and the measurement from Hubble and Gaia collaboration (also known as SH0ES, sometimes miswritten as SHoES, or SHoES).  The problem here is that the error bars no longer overlap, which indicates some sort of problem – either they are measuring different things or at least one of them is measuring the wrong way.

You occasionally read that the Hubble time is a useful estimate of the age of the universe.  In that case, ignoring the ridiculously large error bars on the LIGO/Virgo result, the age of the universe is between 13.01 and 14.60 billion years.  In some cases, it is suggested that the Hubble time indicates how long the universe has been expanding – but to all intents and purposes this is what is meant by the age of the universe (much in the same way as a baby is not strictly 0 days old at birth, usually having gestated in a womb for about 9 months, we just pick a nice convenient reference point and count from there).

However, we are now being told that, about 5 billion years ago, the expansion of the universe started accelerating.  If the Hubble time is a useful estimate of the age of the universe and the age of the universe is what we are being told (13.8 billion years, or near enough), then don’t we have a problem?  We can work out the value of the Hubble parameter at a Hubble time of approximately 8.8 billion years (let’s call it H-5, meaning H at now minus 5 billion years), and it works out to be about 1.5 times that of today – ie about 111.1 km/s/Mpc.  (Perhaps the acceleration started only 4 billion years ago, at the end of the matter-dominated era and the beginning of the dark-energy-dominated era.  The value of the Hubble parameter corresponding with a Hubble time of 9.8 billion years is a bit lower at H-4=99.8 km/s/Mpc, but still about 40% higher than today.)

What is going on here?

Is the Hubble time only coincidentally a good estimate of the age of the universe at the current time (but won’t be in the future and wasn’t in the past)?  This sounds like it would be a fair addition to the list of fine tunings, and that surely can’t be good.

If the Hubble time isn’t generally a good approximate of the age of the universe, then there’s no reason to suggest that it ought to be a good approximate today and maybe the age of the universe is not 13.8 billion years after all.  Not really, there are a multitude of ways in which the age of the universe is measured, so cosmologists don’t simply rely on inversing the Hubble constant (for example, they look at cosmic background radiation fluctuations).

Another possibility is that the Hubble parameter has been tracking the age of the universe faithfully and 5 billion years ago it actually was H-5=111.1 km/s/Mpc.  Would that mean that, since that time, Hubble expansion of the universe has decreased and some other expansion of the universe (due to dark-energy, apparently) has got involved?  If so, it still doesn’t really add up.  We have the Hubble constant because that’s what we measure as the current expansion rate of the universe.  If the Hubble component of that is lower than is required to account for that expansion rate, then H0 < ~70, which increases the estimate for the age of the universe.  Thinking about the notion that the universe has been accelerating in its rate of expansion for the past 5 billion years, let’s say at best case that H-5 was just a bit lower than today, say just under the error bar for LIGO/Virgo at 61 km/s/Mpc.  That would mean that at that time the Hubble time was 16 billion years, and today the universe would be 21 billion years old.  That’s a big error in measurement.  It just doesn’t seem right.

So, a question for the people in the know … if we were around 5 billion years ago and were measuring the Hubble parameter using the rate at which distant galaxies were receding, approximately what result would we have come up with?

Do I have a solution?  Yes, I think I do.  I just don’t quite understand (yet) why most cosmologists would likely tell me I am wrong.