### Article by Michael Burton, Director of the Armagh Observatory and Planetarium

The Hill of Infinity of Armagh, in the Astropark of the Observatory and Planetarium, contains a scale model of the universe, illustrating the place of planet Earth in the immensity of the Cosmos.  How can this possibly be related to understanding the pandemic caused by the spread of the COVID 19 coronavirus around our planet?  It is because the underlying concepts, between way the scale model of the universe has been constructed and of our understanding of how the growth in the number of cases of the coronavirus occurs, are the same.  They are based on the principle of exponential growth.

Let us explain.  We have become familiar during the lockdown with the concept of the R-factor for the coronavirus.  Of the importance of lowering its value if we wish to slow the spread of infection.  The R-factor is a measure of how easily the virus can spread, simplified into a single number.  It represents the number of people that a person already infected with the virus will pass it onto, on average.  Let’s suppose, for an example, that the R-factor is two, and for simplicity that the transmission occurs within a day (the concepts are the same if the transmission time is longer, but it is easier to illustrate if we stick to a day).  Then, after one day, two more people have been infected, making a total of three.  After two days, they in turn each affect two further people, making a total of seven cases.  And so on.  If we keep on multiplying by two and adding, we will find that there are 255 cases by the end of the first week.

In contrast, suppose the R-factor had been kept to one.  Now each person only infects one other.  Then the number of cases increases by one per day.  After a week there are a total of 8 cases.  We show these comparisons in the first set of graphs (Fig. 1), it is easy to see that there are about 30 times as many cases when R=2 as when R=1.  However R=2 might be too high, you say?  But it is easy to calculate the number of cases for whatever the correct value of R is.  So we also show the scenario when R=1.5 here as well.  In this example, after one week the situation is much better, there are five times fewer cases than when R=2.  However, the situation is still six times worse than when R=1. Figure 1. This graph shows the growth in the number of cases occurring during the week following the first case that is recorded, under three different scenarios for the R-factor. These range from R=1 (each infected person affects one other, on average) to R=2 (each person infects two others). After one week, there are thirty times as many people infected in the later scenario. If we had been able to reduce R to 1.5, then the increase in number of infections over the week would have been five times smaller.

This is an example of what mathematicians call exponential growth.  Numbers are small at first but rapidly rise to dramatic numbers.  We illustrate this in the second set of graphs (Fig. 2).  The time axis has been extended from one week to one month after the first case occurs.  Now the plots are dominated by the scenario when the R-factor is 2.  By continuing to double each day we find that there are over 4 billion cases after one month!  It’s impossible to even see how many cases there have been for the other two smaller values for the R-factor, so large is the number cases recorded when R=2.  This is the power of exponential growth!  In practice, the growth cannot keep up at this rate indefinitely.  Ultimately, resources will limit the total number of cases.  For instance, in the spread of a disease you clearly cannot infect more people than the total population that exists! Figure 2. The same set of graphs, but this time showing the total number of cases after one month for the same three scenarios of R-factor (R=1, 1.5 & 2). The difference between the three scenarios is now astronomical – in fact it is impossible to discern from the scale how many times worse the situation is when the R-factor is 2 compared to the other two values. We need to switch the axis that shows the total number of cases from a linear scale to one that can adequately represent what is happening.

However, the key in limiting infection is to slow down the rate of growth so we don’t reach such disastrous levels.  The plot in Figure 2 does not help us to understand what is needed.  We need to replot the graphs in such a way that we can see how the rate of infection grows for each of the values chosen for the R-factor.  So we change the axis showing number of cases from a linear to a logarithmic scale (Fig. 3).  With this new scale the horizontal lines are no longer evenly spaced apart, rather along each line there are ten times as many cases as along the line below it.  This allows us to use a single plot to show the behaviour for one case as well as for a billion.  There is still a dramatic difference seen between the three scenarios.  This time, however, by reading off the scales we can see that when the R-value is 2 it is 30,000 times worse than when R=1.5, and an incredible 100 million times worse than when R=1!  A truly astronomical difference. Figure 3. This graph now shows the same data but with the axis for the number of cases switched to a logarithmic scale from a linear one. For each horizontal line across the page the number of cases is a factor ten higher than in the line underneath it. We can now readily see there is a dramatic difference between the three scenarios for the R-factor. When R=1 the total number of cases remains small, around thirty at the end of the month. It is apparent from the curves that there are about 30,000 times more cases if we had R=1.5 and an astounding 100 million times more cases when R remains as high as 2. These plots also illustrate the concept of flattening the curve by reducing the R-factor to attain the blue curve for R=1, in doing so dramatically lowering the number of cases.

It is also apparent what is meant by the term flattening the curve, as heard repeatably in the daily media briefings from governments.  The curve for R=1 is flat, it is not for R=2.  If we can achieve a flat curve then we can control the spread of the virus, so limiting the number of people who catch it to relatively small numbers that our public health systems can then cope with.

So how does this all relate to astronomy?!  In the Astropark surrounding the Observatory and Planetarium in Armagh there is a scale model of the cosmos that you can walk around, illustrating what the planets, stars and galaxies are. Actually, there are two scale models, one for the Solar System, and one for the Universe itself.  The first model uses a linear scale for its representation, the second a logarithmic scale, just as in the two representations we were using to describe the spread of the coronavirus.  In the case of the cosmos it is distances between the different types of celestial objects that are growing exponentially, not the number of cases, but the principle is the same.

Along the Solar System walkway is laid out a scale model of the Sun and planets, spaced at their right relative separations (Fig. 4).  The Sun is five steps from Earth, representing 150 million kilometres.  While these distances seem vast to us, we can send spacecraft to the outer planets, their journey times are measured in years, they are comprehensible.  A linear scale allows us to represent them, a gentle stroll in the Astropark then lets us visit all the planets. Figure 4. On the Solar System walkway at Armagh, heading away from the Sun, on the journey between Jupiter and Saturn. The size of Solar System has been shrunk so that, for every five steps you take, you travel the distance from Earth to Sun, 150 million kilometres. On this linear scale we can readily traverse from the Sun to Pluto, visiting the eight planets along the way. As in the scenario when the R-factor was 1, we can represent the concepts here using a linear scale.

However, the walk soon becomes impractical should we decide the leave the Solar System and head to the stars.  The nearest star, Proxima Centauri, would be found close to Paris.  Perhaps achievable with determination?  But in order to reach the centre of our Galaxy we would have to walk a distance that is twenty times further away than the Moon, which is clearly not possible!  We have to change scales to describe the cosmos beyond the Solar System.  We move from a linear scale to a logarithmic scale, just as we did when describing the spread of the coronavirus.

This is how the cosmos is represented on the journey up the Hill of Infinity, passing from the Solar System, past the stars and nebulae in our Galaxy, through the realm of the galaxies, to the Big Bang at the formation of the universe itself at its top.  The journey starts at the Hypercube (Fig 5.), where the display illustrates the concept of logarithms and powers of ten.  There are three cubes in the Hypercube – small, medium and large.  The large cube is ten times bigger, in every dimension, than the medium cube.  Similarly when comparing the medium cube to the small.  The principle here is that when walking up the Hill of Infinity a stone band laid in the ground is passed every ten steps.  Each band represents being ten times further away than the previous band.  So one hundred times further away than two bands away, one thousand times further away than three bands ago, etc.  This is the principle of exponential growth being displayed on a logarithmic scale. Figure 5. The Hypercube at the bottom of the Hill of Infinity in Armagh, at the end of the Solar System walkway. We are now transitioning from linear space to logarithmic space. The three cubes within the hypercube represent this concept. Each is ten times larger, in every dimension, than the cube inside it. As we climb up the Hill of Infinity, after every ten steps we cross a series of stones bands set across the path. Each band represents travelling ten times further than the previous band. So 100 times further than two bands ago, 1,000 times further than three bands ago, etc. This is a logarithmic representation of the distance. It is equivalent to the way we showed the increase in the number of infected cases of COVID 19 when the R-factor was greater than one.

We then get a picture of how the cosmos is structured, a sense of the scale of the Solar System and the stars within our Galaxy, and of the scale of the Galaxy within the larger structure of the universe itself (Fig. 6).  All the way to the moment of creation in the Big Bang, which also represents the size of the observable universe, the very greatest distance that it is possible to view to.  Yet, while the concepts here are utterly different to those being used to understand the spread of the coronavirus, the principles used to display the concepts are the same in both, of exponential increase and logarithms to capture all the scales in the problems at hand. Figure 6. As you walk up the Hill of Infinity markers in the ground indicate the astronomical objects passed along the way. We start with the Solar System and its comets, pass through our Galaxy with its stars and nebulae, traverse the realm of the galaxies and finally, at its summit, reach the birth of the Universe in the Big Bang. This echo of this event can be heard in radio waves that have traversed a distance of 14 billion light years to reach us. The logarithmic scale allows us to include all these amazing concepts and phenomena concerning the nature of the cosmos in a single journey up the Hill of Infinity!

However, with the Hill of Infinity there is one final treat when one reaches the very top (Fig. 7).  Here we find a stone circle, a modern version of the kind our Neolithic ancestors used to mark out the passage of the Sun in our skies, used as the first calendars.  Here is a modern version of this calendar, showing where the Sun rises and sets at the equinoxes and solstices, the turning points for the seasons.  As well as linking us from the mythology of the past to the science of today, and all that captures our imaginations about the heavens above.  Finally, we can turn to take in the view of the City of Armagh, its two great cathedrals, and the beautiful green hills that surround. Figure 7. At the top of the Hill of Infinity we find a stone circle, one that links us to our Neolithic ancestors and the stone circles they built into the landscape. These often had celestial alignments towards the rising and setting of the Sun during the course of the year, calendars that marked the course of the seasons. From this modern version of these stone circle calendars on the Hill of Infinity you can view the direction of the rising and setting Sun at the solstices and equinoxes, and connect back to the era when humanity was first wondering about the cosmos and building monuments in awe of it. Finally, to finish our journey, there is a spectacular view to admire of the City of Armagh and its two Cathedrals, while all around are the beautiful green hills of County Armagh.