Yet another home analemma page: the Mack family driveway and windowpane analemma

The sun crosses the sky at a lower elevation in winter than in summer. If you were to take the position of the sun at a fixed (clock) time of the day (say clock noon) throughout the year, you might guess that the sun's path would be a straight line between its low position in winter and high position in summer, and that the sun would retrace the same path on its return in winter. In fact the path is not a straight line, but a figure "8" called an analemma. (http://www.analemma.de/, see defn 2). Here's Dennis DiCicco's famous photo of an analemma (http://www.analemma.de/images/articles/dicicco.jpg) the first analemma photo I know about.

The analemma shape of the sun's path is caused by the earth's orbit being elliptical, rather than circular. The ellipticity of the earth's orbit

All this information was available to me as a high school student, reading the newspaper at breakfast each day looking for the time of set/rise for the sun and planets. Most of this information was also known by the ancient Greeks, who also knew the correct size and shape of the earth and moon. The ancient Greeks used nested epicycles to approximate the planet's and the moon's orbits. That the shape of the orbits were in fact ellipses, was determined by Kepler, using Tycho Brahe's meticulously collected data.

Using the geocentric view of the sun's daily position, the sun is closest to the earth (perihelion) about 5th Jan and so whizzes past us more quickly. By 6 July, the sun is further away and moves more slowly against the fixed stars (and looks a little smaller). Because of the different speed of the sun against the fixed stars during the year, if you look up at the same time (say noon), some days the sun will be fast (by 17 mins in November) and be slow (by 14 mins in February) (see Sundials on the Internet http://www.sundials.co.uk/equation.htm). The difference between true time (as measured by a clock) and the apparent time (as measured by the position of the sun) is called the "equation of time" and is found on a sundial to enable you to calculate clock time from the sun's position.

An article in Sky and Telescope ("The Analemmas of the Planets", David A. Harvey, Flandrau Planetarium, Sky and Telescope, March 1982, p 237-9) showed the analemmas and the calculations for the other planets. Since the mid '90's, you've been able to draw analemmas on your azimuthal equidistant projection maps (http://www.wm7d.net/az_proj/az_html/az_help.shtml#analemma). Now everyone is doing it e.g. Astrophotography by Anthony Ayiomamitis (http://www.perseus.gr/Astro-Solar-Analemma.htm) and there is hardly a place on earth that hasn't been analemmaed.

The analemma is the path that the sun would take through the sky if the earth rotated about its axis only once a year (i.e. kept the same side pointed towards the sun). Because the earth rotates every 24hrs, rather than once a year, to see the analemma, we have to look at the sun at the same time every day. Because the moon keeps the same face to the earth, astronauts on the moon looking up at the earth would see it move through an analemma shaped path every month.

In the times not so long past, an understanding of the analemma was so basic that they were drawn on globes of the earth. The analemma rated up with the equator and the poles. By the time I entered grade school, the dark ages of education were upon us, and teachers mumbled incoherently when asked to explain the analemma. The embarrassement of the teacher's on discovering their ignorance of the motion of the sun, at the level understood by the ancient Greeks, was handled deftly by globemakers who, anxious to retain their customers, removed the analemma from globes ^[1]. We are now in an era where graduates of the education system leave with a view of the solar system that predates the ancient Greeks. This is a fitting educational outcome in a country where both major political parties agree that our best exports are jobs, subsidised agricultural goods and promisory notes payable by our children.

Growing up in the southern hemisphere and being being subject to the Euro/American northern hemispherism of our education system, gave us plenty of opportunity for testing our knowledge. To navigate by the stars, you look for Polaris, which as everyone (well the Scout manuals) knows is directly north ^[2] . The books talked about 4 seasons: summer, winter, spring and fall - if you can believe that. Unbeknown to the small fraction of the world's population in USA and Europe, most of the world has wet and dry, or dry and dry, or wet and wet. Where I grew up, the climate was dominated by the southern oscillations: we had dry and occasionally wet and when it was wet, there was lots of it, but only once every 10-20 yrs. Sure, sometimes wet comes with cold and sometimes it comes with hot, but plants and animals breed and grow on water, not temperature. An agriculture, based on crops and animals that couldn't survive 20yrs of drought, was doomed from the start. It was only the willingness of the Australian taxpayer to compensate farmers for decades of "unexpected" crop losses that kept the system functioning for 200 yrs. Sundials are latitude dependant: in particular in the southern hemisphere the sun crosses the sky from the right to the left (anticlockwise, the opposite direction to the northern hemisphere). When I grew up in Australia, in the '50's and 60's, no-one would be caught dead buying anything made in Australia. The sales clerk would only have to say to my mother "but Modom, this is British" and that would settle it. The article would be snapped up, the purchase assuring the sales clerk that my mother recognised only the finer things of life. The problem with a sundial made for a latitude of 60N, sitting in a garden at 34S, was that it would give a time in the morning when it was the afternoon. Parks and gardens were filled with transplanted sundials, installed by credulous burghers wishing to simultaneously declare their love of art and pay homage to the scientific accomplishments of the ancients. Running the gauntlet of the Australian education system was a punishing proving ground for the critical thinking of a young person.

Although my azimuthal equidistant projection map generator (http://www.wm7d.net/az_proj/az_html/azproj_help.shtml#analemma) has included an analemma since the mid '90's, I didn't know about analemmic sundials. I occasionally visit my native Sydney, Australia to see friends and to walk in the Blue Mountains. On returning from one hike, I bought fish and chips in Richmond, on the banks of the Hawkesbury River. Looking for a place to eat, the proprieter gave me directions to a local park. He said it was a little difficult to find, but to not be discouraged. It wasn't far, but there were no signs to it and I had the place to myself on the banks of the river at sunset at a time when the place should have been jumping with people. Why do they have this nice park if no-one goes there? I guess Australia is such a nice place, that you don't need parks to enjoy the outdoors :-)

There to my surprise, on the edge of the river, was an analemmic sundial.

Figure 2. Analemma in park in Richmond - photo courtesy Janelle Hatherly

Figure 3. Analemma in park in Richmond - photo courtesy Janelle Hatherly

An analemma, on the ground, was marked by the date, and was surrounded by an arc of stones marking the hour. A person (of any height) stood on the analemma at the spot for the day and looked at their shadow on the arc of stones. It was apparent that the analemma automatically corrected for the equation of time. I hadn't realised this was possible. I wondered if it was an exact solution or only approximate. Did it work all day, all year? What was the shape of the arc of stones? I did some calculations on my return to USA, but wasn't convinced that I'd done them correctly.

On my next trip back to Sydney, I visited Max Hatherly, a friend's father (who lived in Stannix Park near the park with the analemmic sundial) and mentioned the analemmic sundial, how pleased I was to find it, and that I was amazed such a thing existed. He grinned and said "I designed that sundial" and pulled out the local phone book from a few years before.

Figure 4. Phonebook celebrating the start of work on the Richmond Analemma - photo courtesy Janelle Hatherly

The front cover showed Max (on the left) and the team that build the analemmic sundial, with each person standing behind one of the hour marker stones. Max then took me to his garden to see his personal analemmic sundial.

Figure 5. Max Hatherly at the Stannix Park personal analemmic sundial, with the author in 2009, many years later.

Figure 6. Max Hatherly at the Stannix Park personal analemma, 2009.

Note

Note that the summer lobe of the southern hemisphere analemma (where Max is standing) is the larger. Compare this lobe with the small summer lobe of the North Carolina (northern hemisphere) analemma (see taking a reading). The relative size of the two lobes depends on the plane in which the analemma is projected. While the summer lobe (closest to the gnomon) of the NC analemma projected on the ground is the smaller, the summer lobe (the upper lobe) of the NC analemma projected onto the vertical glass door is the larger (see front door analemma from outside). The door in NC, being vertical, would be in the horizontal plane if moved to 90-38=52°S i.e. the door analemma in NC looks like a ground analemma in the southern hemisphere. The spacing of consecutive dots in the summer lobe projected on the ground is small, while the spacing of consecutive dots of the summer lobe projected on the vertical glass door is large.

I gathered that Max had done more than just design the park's sundial; he may have proposed and helped raise the funds too.

Many months later, after my return to USA, I remembered that on a previous trip, I'd told Max I was going to visit the Mt Annan Botanic Garden (http://www.rbgsys.gov.au/mount_annan_botanic_garden/). He'd smiled and said "make sure you look at the analemmic sundial". Back then, I didn't think it anything special - it was just another sundial. I'd missed the point.

If you can watch the analemma from all of the ancient temples in Greece, then it could be done from the driveway of my house in Durham, North Carolina, USA. What school nowadays can be without an analemmic sundial? The science teacher at my son's school was interested, so we looked for ways to put an analemma in the school too.

Here's the spray painted dots.

Figure 7. Analemma spray painted dots looking south.

The spots closest to you were done 9 months earlier and are a bit faded.

Figure 8. Analemma spray painted dots looking north.

The spots closest to you were done 3 months earlier and are relatively bright.

The gap on the left side, below the cross over point, occurred during our summer vacation - we were away for 2 weekends. Note the miss-aligned point on the right side of the lower loop - I did that one myself - so I can't blame the family, I have no idea what went wrong there. There is an inflexion on the lower left side with a concave section. This shouldn't be there. In general the analemma is not as smooth as I'd hoped :-(

Figure 9. Taking a reading.

The cups indicate previous readings. The red cups are the noon analemma.

My son in the foreground is holding the gnomon. The 3-axis bubble level is the yellow device 1/3 the way up the gnomon. I'm in the back holding the marking chalk in my right hand. The can of spray paint is on the ground to my right, mostly occluded by the gnomon. The camera is looking north (east is to the right of the picture). The yellow cups are marks taken in the morning, the blue cups are taken in the afternoon. This photo was staged, and was taken about an hour before local mean noon (note: the shadow of the pole passes the nearest yellow cup and not the red cups as it did an hour later; and both our shadows are not parallel to the axis of the gnomon).

	Note
	The summer lobe of the nothern hemisphere analemma (closest to my son holding the gnomon) is the smaller lobe, whereas in the southern hemisphere, the summer lobe is the larger lobe.

Figure 10. A reading close up:

spray paint; the cardboard mask for spraying (initially the large hole with a diameter of about 2mins was used; later when accuracy became a goal, a much smaller hole was use); the gnomon (pole on right, laid on ground) with 3 axis bubble level (vertical bubble not used, the two horizontal ones at the top of the yellow frame are used). The bubble level is attached with silicon glue - the rope is just incase the glue comes undone. A line of pink chalk marks are across the image below the cardboard, and opposite the yellow bubble level frame. The driveway concrete is a mixture of mafic (dark) and felsic (light) material.

The photo here shows about 15mins of chalk marks (seen better in the enlarged photo). on a partially cloudy day It's Sep 24 (northern hemisphere fall), just after the sun has crossed the analemma's node. The image is looking away from the sun (north, towards the winter loop of the analemma). Local mean noon (due to daylight savings) was at 1:16pm. The data (a line of pink chalk marks across the image from left to right, towards the yellow bubble level on the gnomon) start about 1:00pm at 1 minute intervals. There is a gap a 1:07 and 1:08pm and another for 1:16 and 1:17pm, due to cloud. The position of the sun at 1:16 is interpolated from the position at 1:15 and 1:18pm. The calculated position at 1:16 is marked by pink chalk lines above and below the calculated position. The radial jitter in the line corresponds to about 15secs of time error (my initial reproducability estimate). This position of the mark differs from last year's by by about 1/2 a shadow diameter (if they were both good readings, they'd be on the same line). Presumably the mark I collected last year (at the beginning of the project) was not a good one. The new point, with 15mins of pre-data, fits the overall curve better than last year's point.

Figure 11. Looking east.

The E-W arcs of cups are the shadows at 30min intervals during the day. The yellow cups are the marks taken before noon, the blue cups are the marks taken after noon. The red cups are the noon analemma.

The analemma was started in Sep 2004 after returning from our summer vacation. It didn't occur to me till May 2005 to take marks at any other time than noon. This was about the time when the sun is at the analemma's node. The three E-W arcs are taken (L-to-R) about 3 days before the fall solstice (Sep 2005), the (May 2005) analemma node and the summer solstice (Jun 2005). Note that the equinox line is (almost) straight, while the other two lines are increasingly hyperbolic.

Although I didn't do this, it should be possible to determine the day when the locus of the sun's shadow is a straight line (occurs at the equinox). I got within a day of the date of the equinox with the door analemma (see the equinox as seen by the door analemma).

I did 3 more arcs (hyperbolas) on the winter side, one in the late fall, one at the winter soltice and one in the early spring, when the sun was about the same elevation as in the fall.

Figure 12. Looking south east. Red cups are the path of the sun through the day at 30 min intervals. Yellow cups are on the analemma.

Misconception: local noon is not the same as local mean noon

I'm embarrassed to say that even though I know the definition of these two terms, when it came to applying them, I got it wrong.

• local noon:

  (see Noon http://en.wikipedia.org/wiki/Noon). Meridians are imaginary lines in the sky joining the north pole of the sky with the south pole of the sky. The zenith is the point directly overhead (if you held a plumb bob on a string, the string would point to the zenith). The meridian line that passes through the zenith is officially called the "Celestial Meridian", but usually just called "the meridian" (see The Celestial Meridian and Zenith http://www.herongyang.com/astrology_horoscope/Astronomy_The_Celestial_Meridian_and_Zenith.html). When the sun reaches the meridian, it's (local or solar) noon. Shadows are shortest; the elevation of the sun is maximum.

  If you don't know about watches or time, local noon is a simple concept: you look up in the sky - if the sun is at it's highest elevation, then it's (local) noon.

  Since the earth's orbit is almost circular and the line of apsides (see Apsis http://en.wikipedia.org/wiki/Apsis) (the line joining the perihelion point to the aphelion point) is almost colinear with the line joining the solstices, the concept of solar time being constant (regular) works quite well.

• local mean noon:

  If you measure time by a clock, then because the earth's orbit is elliptical, the length of the days (local noon to local noon) is different each day. Some days are 30secs longer than the average, some 30secs shorter than average. These differences accumulate and consequently local noon does not occur at the same clock time each day, varying by 14+17=31 mins through the year.

  For convenience, we define clock time in terms of the rotation of the earth with respect to fixed stars (stars that are so far away that they don't move) and not with respect to the sun. The local mean noon then is the average of the noons over all the year (I can be more exact about this, but it would take a lot of words).

	Note
	For convenience, all people in the same time zone set their watches to the time at the standard meridian in that timezone (usually a multiple of 15° from the Prime Meridian). So when it's noon on your watch, it's local mean noon at the standard meridian.

Pretty simple huh? OK quiz time:

• If you marked the shadow of a gnomon at local noon each day, what shape would you see on the ground?
• What would you see if you did it at local mean noon?
• On what line(s) lies the maxima/minima of the hyperbolas? (the hyperbolas being the locus of the shadow of the sun's path through a particular day)

(answer: ^[4] .) If you got the right answers, skip past the next part of the discussion to the following section.

I don't know what I must have been thinking, but I assumed that the bottom of the hyperbolas (seen in photo above) was on the analemma. As I found out the bottom of the analemmas is on the projection of the meridian, a straight line down the middle of the analemma. I picked dates in the spring and fall to do the hyperbolas, with the widest separation between the ascending and descending sides of the winter lobe of the analemma. I thought that the different locations of the minima of the two hyperbolas would be clearest at these two points. As I found on the day, the two hyperbolas were parallel the whole way. The stationary point on the hyperbolas (when the shadows are shortest) occurs at local noon. I was mortified to realise my misunderstanding.

Figure 13. Looking north.

This is the same scene as immediately above, except looking north.

Although teachers are glad of interested parents, I found the administration less welcoming. I had previously tried to introduce the National Archery in the Schools program into the school, for those kids who didn't want the standard sports. This is competitive archery using Olympic competition rules. I'd arranged for the equipment to be sponsored (about $2,500) and for certification of myself as a coach (I'm also a professional soccer referee). Although there's never been an accident in this program, I arranged for $1,000,000 insurance for each kid. The cost to the school? - $0. All I needed was a field or gym to do it. The school wasn't interested; no explanation was given or thought neccessary. It took multiple calls to even get a reply to my proposal. I'm a member of the Carolina Geological Society. In 2008 we visited the Lee Creek phosphate mine in Aurora NC, world's best site for pliocene marine mammal fossils. There were whale parts (and shark teeth) on the ground for the taking. Outside the Aurora Fossil Museum is a large sand box of the same material, whose contents are refreshed regularly by a dump truck from the mine, for kids to riffle through. A whole bunch of kids were having a whale of time going through it the day we were there. The mine will happily transport (I believe at no cost) a dump truck of the same stuff to any interested high school. I brought back a supply of spare whale parts, shark teeth and phosphate ore from the mine for the school's science teacher. The phosphate ore has been sitting in waterlogged sand for 20My and is as insoluble as a rock could be. The teacher already knew about the dump truck offer. The school had turned her down; phosphate was a major pollutant and they weren't going to turn the school into a superfund site just for the education of students.

Collecting data for the analemma was problematic. You're on someone else's turf and obeying the rules of people, who have higher priorities than helping kids tackle large problems with simple tools. I had hoped that the analemma would be constructed dot-by-dot on the playground blacktop, where the kids would see the analemma developing and maybe ask questions about it. Having the kids mark the spots would increase their involvement, otherwise the project would just be another of the things that adults produce out of the blue to take up the kid's time. However kids aren't allowed to duck out of the classroom unsupervised - they might be abducted (or run off). I had hoped that I'd get some photos of the kids with the analemma, but I'd need a legal release from each parent and the school.

The first year, a request by the science teacher to mark the playground blacktop was greeted with "you're not to deface the school property" and "if the school board wanted you to have an analemma, you would already have one". The school doesn't have an analemma, but it does have a sundial mounted prominently above the south facing entrance. Everything in a school is for educational purposes, right? You'd be run down by a bus if you tried to view the sundial; the science teacher wasn't consulted on its purchase, so it wasn't a planned part of the curriculum; there is no correction for the equation of time. Except for October, when it gives correct time, the sundial is off by upto 30minutes. Presumably only the kids have noticed.

kid: Mommy, what's that?

Mommy: It's a sundial, one of the first scientific instruments. It's been used for thousands of years to tell the time. The daily passage of the sun through the sky, marked by the shadow of a stick on the ground, is so regular, that it was the original definition of time.

kid: so how come it's off by 30mins?

Mommy: It isn't a real sundial. It's art. It's to show you that there are more important things to learn at school than your teacher's lessons.

kid: like what?

Mommy: Like when you spend other people's money, you aren't accountable for spending it well. Watch out for the bus.

Since the school is 16 minutes offset from the standard meridian, it's likely that the sundial is displaying local time rather than standard time. This is a reasonable thing for a sundial to do, but it's one step more complicated to explain than a sundial displaying standard time. The sundial should overlook the playground, not a spot where a kid risks instant death. It's unfortunate that the administration doesn't recognise the educational value of a sundial and installed it as a non-functional piece of art. It seems that sundials continue to be a universal tool for testing a kid's critical thinking.

That was the first year.

The next year the science teacher secretly marked the driveway infront of the school, where the kids are picked up, using the school flagpole as a gnomon. Black marker pen on blacktop looked like oil drops. No-one at the school knew what was happening (even the kids), but if you knew what to look for, you could see a nice analemma developing through the year. I was reminded why secret societies and martial arts like karate (empty hand) existed: to propagate knowledge when rulers controlled the masses by keeping them ignorant and powerless. You encode your knowledge in pictures on cards and tell everyone that it's a game.

When classes started the next year and with the pattern revealed, some of the kids would start asking questions. But after a year of data collection and with a nice analemma displayed for all to see, the school repainted the driveway during the summer break, getting rid of all those nasty oil marks.

That was the 2nd year.

We've just started (at the summer solstice) on the 3rd year, I pushed golf tees into the ground on an otherwise unused piece of ground infront of the school, using streetlights as gnomons. The ground is like concrete and I use a pin-driver and hammer to knock a hole in the ground before whacking the golf tee in the last 1/4inch. The ground's people were informed of the golf tees, flush to the ground, but were concerned that they'd damage the mowers. One day I came along to find many of the tees ripped out. Then later after inserting about 100 golf tees, I came along to see that the ground had been dug up with an aerator and only a few tees were left in the ground.

I gave up. 3 years is enough.

It seems that the teachers' idea of an intersted parent and the administration's idea are quite different. The administration's idea of interest appears to be donations of money without any strings attached. If you want to know why american education is lagging, you only have to look at the treatment interested parents and enthusiastic teachers get. How much harder must it be for the students?

In my search for a better gnomon and for a solution that could possibly be used in a school room, one day I realised that the gnomon could move, while the shadow spot remained stationary. The front door on our house has a large glass pane, though which the sun shone onto a wooden floor. The floor showed enough features, that I could use one as the shadow spot, without having to mark the floor (I used masking tape to mark the spot in case I forgot).

Figure 14. front door analemma with the author

The front door faces S.E., so the sun's path across the sky doesn't illuminate the door symmetrically with respect to noon (i.e. morning and afternoon are not the same). Since I was already busy at local mean noon (12:16pm), I marked the front door at local noon (12:00) whenever I was home. After a year or so, I decided that I'd also mark the door at 30min intervals, on the weekend (if I was at home and the weather cooperated) and I'd join the line of dots for every 4th week.

	Note
	The spacing of consecutive dots in the summer lobe projected on the ground is small, while the spacing of consecutive dots of the summer lobe projected on the vertical glass door is large.

I initially marked the glass with marker pen, but the dots faded from the sun in a few weeks. I changed to house brand colored peel-off strips of sticky dots from the local office supply store. They only had one color, and soon I needed more colors (for weekends and different years). Looking elsewhere, I then wound up with dots that came off easily. The front door in the morning has people pushing against it with bags and gear and it wasn't long before the dots were on the floor. I then found "Maco Color Coding Labels", 1/4" diam in reals of 450 dots, numbered MR404-x where x is different for red, yellow, green and blue (from Litonlabels). These have worked just fine. Be careful; these dots come in "permanent" and "removable" adhesive varieties. Make sure you get the permanent ones (the removable ones don't stick to much of anything that I could find).

I started off a window pane analemma in the science teacher's class room, but her window is shaded most of the year, and she hasn't continued marking dots. I expect a futile year of marking the school's driveway broke her spirit.

Taking a photo of a transparent window is harder than I thought. You wind up with a photo of reflections of the neighbour's house or your living room (depending on the location of the camera). The only way out of this seems to be to take a photo at night and with the lights off. If you use a flash (and you don't have a remote flash), then you get a photo of a flash. Here's my best attempt, illuminating with incandescent lights from the side. The dots are only colored facing inside the house.

Figure 15. front door analemma from the outside

• Data is read on the hour and half-hour. Note that the most dense analemma (taken at local noon) is not vertical; it's taken 16 minutes before local mean noon.
• The analemmas left to right are
  • 1pm (goes off the door at bottom left in summer)
  • 12:30pm (top is truncated by overhanging eve in summer)
  • 12:00 (leans slightly to right, is the most dense analemma)
  • 11:30 and 11:00 are complete;
  • Earlier times are progressively more truncated. There is only one point for 8:30am in the late winter.
• At the bottom of the door, the analemmas overlap, making disentangling the analemmas a bit hard from the outside (on the inside, the analemmas are color coded).
• The lines running top-left to bottom-right are the locus of the eclipsing point about every 4th week. Because the door faces SE, these lines are not symmetrical about a horizontal line in the middle of the door.
• The distance from the door to the mark on the floor is too small to see differences in the sun's position of less than 1 min.

Figure 16. front door analemma from the inside

• Data is read on the hour and half-hour. Note that the most dense analemma (taken at local noon) is not vertical; it's taken 16 minutes before local mean noon.
• color scheme is lost in the depths of time. Most recently, yellow (in this photo seen as white) is the descending arm; red and green are ascending arm from last year and this year respectively (to differentiate the analemmas when they overlap).
• The analemmas right to left (going in the reverse order to the image from the outside so that I can use the same instructions)
  • 1pm (goes off the door at bottom left in summer)
  • 12:30pm (top is truncated by overhanging eve in summer)
  • 12:00 (leans slightly to right, is the most dense analemma)
  • 11:30 and 11:00 are complete;
  • Earlier times are progressively more truncated. There is only one point for 8:30am in the late winter.
• At the bottom of the door, the analemmas overlap. Hopefully the different colors help differentiate the ascending and descending arms of the analemmas.
• The lines running bottom-left to top-right are the locus of the eclipsing point about every 4th week. Because the door faces SE, these lines are not symmetrical about a horizontal line in the middle of the door.
• The distance from the door to the mark on the floor is too small to see differences in the sun's position of less than 1 min.

The day of the equinox (+/-1 day) can be seen on the door analemma.

	Note
	You can't tell it's the equinox from the door, but you can tell that the locus of the shadow of the sun is a straight line. If you were living in Greece 2000 yrs ago, you would have wondered what it meant.

There is a red line (concave down) starting at the bottom left of the analemma as seen from the inside. There is a band of glass with no red lines above it, and then another red line (concave up). Somewhere in this band would go a line of dots that's straight. In the middle of the band is 3 series of closely spaced dots. The top red series is from a previous year and should be ignored. The two sets of green dots below it are two days apart. The lower one is taken the day before the spring equinox in 2009. I just collected this data routinely and it wasn't till late the next day (the equinox) that I discovered that I could see curvature on the previous day's reading (I opened the door an looked at the line from the side of the door). By then it was too late to take a set of readings on the equinox, but the day after I took another set of data (now one set the day before and one set the day after the equinox). The 2nd set was concave upwards. When facing the door, these two lines can be seen to diverge in the rightwards direction. Presumably if I'd taken a line on the equinox, it would have been straight. Presumably if I'd taken a line on the equinox, it would have been straight..

By the the planning of Pope Gregory XIII (http://www.geocities.com/calendopaedia/gregory.htm), the day of the week (e.g. a saturday) advances by 1 (or 2 for a leap year) days each year (i.e. if a particular saturday was the 13th day of the year, then the next year it would be the 14th day of the year). I thought that if I kept collecting data on saturdays, I would get new points each year, till the 5th (or maybe 6th) year when I'll be get to redo my original points again and I would have a completely filled in analemma.

Initially taking a reading was a two person job; one to keep the gnomon vertical (possibly in the breeze); the other to mark the shadow. Often there were manpower problems on the day, so a one-person method was developed. The chalk was put on the ground near the last shadow spot (as a coarse reference), you'd hold the gnomon vertical, then in turn look at the bubble levels, then hoping you didn't move the gnomon, look at the clock and the shadow on the ground, scanning for a convenient feature in the random noise of gravel in the concrete driveway to act as a reference. Since the shadow didn't move all that fast, you had a pretty good idea where it would be at the exact minute. When the moment came, you'd walk over keeping your eye on the reference spot and mark it with the chalk. The shadow of the tennis ball took about 3 mins to pass over a spot on the ground. When I later changed to the WWVB clock with a seconds display, I held the clock next to the bubble levels and only looked up to the shadow on the ground at the exact minute. This method gave a lot less scatter.

Initially I just took one reading at the exact time. If the moment was cloudy, I'd just forget about reading that day. It didn't take much thinking to realise that you can't throw away a good reading, just because it's cloudy - you won't get that data point again for another year. I started taking one minute interval marks leading up to the exact moment, and depending on the cloud cover, might have to interpolate or extrapolate to the required moment. I expected a nice straight line with 15secs of scatter, but found that the reproducability was a lot worse than my initial tests. The errors didn't account for some of the really bad points, but were about the right size (1 min of time error) for many of the bad points. I then decided to take about 5 mins of reading for any one mark.

The initial (before WWVB) clock setting error was 15secs. Initial reproducability tests gave a positioning accuracy corresponding to about 15-30secs of time error (about 1/8th the diameter of the shadow). The actual analemma as it developed over the months was much lumpier than expected, with an error corresponding to about 1min of time error - about 1/2 the diameter of the shadow. As well there were the occasional obvious outliers.

While I can attribute bad data at the beginning of the project, to lack of practice, even after 9 months of collecting data, I still was getting bad data e.g. Analemma spray painted dots looking north. "> data taken at the summer solstice (there are a couple of spots not in the line, one on the right about 4 dots from the summer solstice and one at the bottom left just after the summer solstice - the loop on the left side should be convex and not have a concave section).

With 30 yrs of experimental science under my belt, including a paper on estimating errors, I was greatly chagrined to find that the experimental errors were much worse than I'd estimated and I didn't really know what caused them. I have these clues