This is a design for analemmatic sundials, large or small, with elliptical dials. The one I made with my son is twelve feet across, drawn in sidewalk chalk on a level part of our driveway, and you check the time by standing in the spot corresponding to the date and seeing where the middle of your ow shadow falls. You can make a permanent one by first drawing in chalk and then painting over it (but we are renting the house).

Drawing these is fun, and makes a good science education project for small or large kids (one's own, or a class at school). It should take a little over an hour to do the basic measurements and drawing, more if one wants it to be more decorative. Depending on the size and the features of your location, you may need up to two people's help with drawing the ellipse, though I will also say a little about how you can make it without drawing an ellipse.

Analemmatic sundials have a vertical gnomon (the pointer that makes a shadow) that is placed in a different location depending on the date, because the sun's position in the sky differs from month to month. They are often made with the user acting as the gnomon, and the user's shadow showing the time. The design is elliptical in shape and needs to have its dimensions and layout carefully customized for your latitude, longitude and timezone. The design does not work well for latitudes within ten degrees of the equator.

Most of my time on the project was spent writing a sundial generator script to compute the dimensions and layout for the sundial for a given latitude, longitude and timezone. But you don't have to do that. You can now just go to my Analemmatic Sundial Generator website, input your location data, and the generator will automatically produce a PDF file you can print out that has all of the dimensions you need to measure out and draw the sundial. And if you want to make a small sundial (e.g., on a piece of wood or metal, with holes drilled for the movable gnomon), you can just trace the drawing from the PDF file. The script is open source, so you can download it and modify it for your own purposes.

The theory of analemmatic sundials is nicely explained here in mathematical detail (I got most of the formulas for my generator from there). You can also see the Wikipedia article for photos that might inspire you.

In the following instructions, I will assume that like me you will make the design in chalk on a large, level paved area. Please make sure you have the legal right to draw in chalk where you are doing this. This would be a great project on a college campus if there is an area where chalk drawing is allowed (or you can get permission from the administration), or else you can do it on your own driveway or a walkway if it's level. If you do it at home, you can make it permanent by painting over it.

Many variations are possible. Chalk on concrete or asphalt is not the only way. You can draw it with a stick in soil or sand or snow, for a very impermanent one. Or you could also lay it out on grass (see the Step 4 variations), and then put flagstones--or sticks, for a temporary version while camping--where the hours are, or you could just make a small one out of wood or metal.

Tools needed:

measuring tape and/or yardstick (both is better)
unstretchy string (my driveway sundial needed about 22 feet of string)
something with a right angle (e.g., a piece of paper, a box, a carpenter's square, etc.)
chalk (I recommend ordinary board chalk for the initial drawing, and then going over with either paint or sidewalk chalk for final lines)
enough flat and level space for the sundial (12 feet across is good for a child-sized one; proportionately more for adults; less if you want to hold a stick in place as a gnomon)
two pencils / screwdrivers / pieces of dowel / etc.
computer with Internet access and PDF viewer
printer and paper (you could also take a laptop outside without printing, but you'll have a hard time keeping it free from chalk dust)
optional: paint
optional: magnetic compass

As a warmup, you might consider doing this even simpler non-analemmatic paper sundial.

Step 1: Running the Sundial Generator

You will need to have the following information:

your zip code (if you're in the US) or latitude/longitude (this site may help)
your timezone
whether your location has daylight savings time (time change between winter and summer time).

You will also need to decide how wide your sundial is going to be (mine is about 12 feet or 360cm; it's probably fine for someone a bit more than five feet tall), and what you want to do about daylight savings time. If your location has daylight savings, you have two main choices: label the clock with winter time and then add an hour to what the clock shows during summer time, or label the clock with summer time and then subtract an hour during winter time. If you are making a temporary installation, you can also just use the current daylight setting, and not worry about what will happen after the time change.

Now go to my script's website analemmatic.sf.net. Start by filling out your desired actual sundial width, and select the units. The units you choose will determine the units in which all your dimensions will be calculated. If you choose millimeters, centimeters or inches, the script will give you all your dimensions in these units. (Note: If you chose inches, and your sundial is not too large, your dimensions will include fractional parts, e.g., 2 7/8". Of course, decimal is used with metric units.) If you choose feet, the script will give you all your dimensions in feet and inches (and fractional parts of inches if the sundial is not too large).

Then fill out your zip code or latitude/longitude, choose your timezone, and select what you want to do about daylight savings.

I recommend you leave the "Include (x,y) coordinates" option unchecked. The one exception is if you are unable to draw the ellipse on your surface (e.g., you're putting down flagstones on grass) and want to use an alternate method that does not involve drawing an ellipse, but measuring out hour marks with two coordinates (see the second half of Step 4). Please leave the "dimensions and instructions" option checked, unless you just want to print out a scale drawing of the final sundial with no dimensions (which might be fun if you want a paper sundial).

Click on "Submit" to generate your sundial PDF. If you are having trouble viewing it, you may need to first download a PDF viewer here and/or try a different browser. Print out the file. You will now have pages corresponding to steps 2-6 in this Instructable.

Note 1: Although I will give excruciating detail of the steps, you might just find the PDF file self-explanatory.

Note 2: The instructions will include images of the PDF file for my location. Yours will look somewhat different (especially different if you're in the southern hemisphere). Do not use my images--run the generator for your own. The exact shape ("eccentricity") of the ellipse matters crucially for this design of sundial.

Step 2: Draw the Axes

In this step you will mark the major (longer, horizontal line in the PDF file) and minor (shorter, vertical line) axes. These can be erased when you are done with the sundial.

I recommend chalking out the sundial with blackboard chalk, as blackboard chalk can be very easily erased with a wet rag even on quite rough pavement.

The horizontal line (I will talk of "vertical" and "horizontal" with the PDF file held the normal way so that the text faces the way they normally do) needs to run east-west. The vertical line needs to run north-south. Each page on the PDF file has an arrow showing which way north should go.

You will need to find out which direction true geographic (not magnetic) north is. There are three ways of doing that. The first way is the simplest but less precise.

1. Take the last page of your PDF file which shows the completed sundial. Put it on level ground where it is sunlit. Place a pencil upright on the vertical line on your correct date (e.g., if it's August 10, put it in the August area, a little closer to the July end). Check what time it actually is. Rotate the sundial, keeping the pencil in place, until the shadow of the pencil shows the correct time. (See first photo.) The "N" arrow on the page will then point north. (For slightly higher precision, use the time correction chart--see Step 7 for how it works.)

2. Use a magnetic compass, but you must correct for magnetic declination (the geographic north pole is not the same as the magnetic north pole) or your sundial will show the wrong time. Read this Wikipedia article and follow the links at the bottom of it to find out what the magnetic declination at your location is. If your declination is, say +10° (sometimes shown as "10° E"), true north will be 10 degrees to the west of your compass's north needle (the "E" in "10° E" means that magnetic north is east of true north); if your declination is -10° (or "10° W"), true north will be 10 degrees to the east of your compass's north needle.

3. At night, find the north star, and carefully draw an arrow on the ground aligned with it (e.g., by setting up a telescope or using a quadrant).

Also, a helpful hint is that in many cities streets and house walls are aligned along true geographic directions.

Once you know which way is north, draw a center point for the two axes, and then draw axes outward from the center point (towards the north, south, east and west), using the dimensions shown on Step 2 of the PDF file. Make sure the two axes meet at right angles. I used a piece of shelving I had in the garage for making things be at right angles(photo 2).

Hint: If you can find a joint between paving stones that runs exactly east-west, using that for your east west line will make drawing the ellipse a lot easier. In this case, you don't need to draw the horizontal line at all, but you do need to measure off and mark its center and endpoints. If the joint has significant width (more than about an 1/8" or 0.5cm), you will need to decide which edge of the joint counts as your official line, and do all your up/down measurements from that.

Step 3: Draw the Ellipse

An ellipse can be drawn by fixing two pins, putting a loop of string around them, and then using some chalk within the loop of string to draw the ellipse, going around the pins while keeping the string taut. Each pin is a focus of the ellipse.

If it turns out that you absolutely can't draw an ellipse, go to the second or third part of Step 4 instead. But I strongly recommend drawing the ellipse. It's not so hard, unless your surface doesn't allow it.

The Step 3 page of your PDF file will tell you how large a loop of string you need for your location and dimensions. Choose string that stretches as little as possible. I used kite string, though I would have preferred something with a little less stretch. Carefully measure out the length indicated in the PDF file, making sure the string is taut while it's being measured, and tie the loop closed. Make sure the loop has the length indicated so don't tie it any shorter than the length you measured out. (The distance in the PDF is the distance all the way around the loop.) Make sure that the knot won't slip. (If you're not good at knots or your string is slippery, you can put a drop of superglue on the knot.)

The Step 3 page of the PDF also shows where the foci are. The locations of the foci are indicated by small circles on the horizontal (east-west) line, and their distance from the center is shown. Measure out these distances on the horizontal line from the center, and mark the points. Now you need to set up pins. If you were lucky enough to set up the east-west line along a joint in the pavement, you can just stick the pins into the joint (if the joint is wide, stick them into one side, and make sure that's the side you take your later measurements from). I used screwdrivers as my pins. Make sure the pins are upright.

If you don't have a line in the pavement, and there are no cracks you can slip a nail into right where you need it, you will need two people to help you. Each helper will hold one pin upright in the correct location, making sure it stays in its correct location. I think small kids might not be the best at this--it might be better if adults hold the pins and kids do the ellipse drawing. You can use all sorts of pins. E.g., a pencil with the eraser on the ground to keep it from sliding around, or a piece of 1/4" dowel. Depending on the lengths of the helper's arms and the dimensions of the sundial, a single helper might hold two pins, but that's unlikely. Make sure the pin stays upright.

If you can't get helpers and can't find a crack and own the piece of pavement, I suppose you can use a thin masonry bit to drill a hole and then put in a long nail. That's at your own risk. Better to find helpers.

Once you have the pins, put the loop of string around them, put chalk within the loop, and make a taut triangle, but not so tight that it makes the pins lean. You can now double check the dimensions of the string and the axes, by checking the string just allows the chalk to touch all four ends of axes.

Then carefully draw the ellipse, looping around the pins. (If you have helpers, you may have to stop from time to time so they can move out of the way of the string and put the pins back.) If you make the pins lean over, the ellipse will be lopsided. That happened to us. But just straighten the pins, go over the bad segments again correctly, and when you're all done, erase the mistakes with a wet rag. The second photo shows such mistakes, but they were easy to erase.

This is the most fun step I think. There is something magical about watching the ellipse form.

If you don't want to, you don't have to draw the whole ellipse, but only the half (or a little more than half) that covers the times you want to show on the clock (see the next step on the PDF).

Educational suggestions: If you're doing this with kids, take the opportunity to ask them to imagine what shape you'd get if the two pins were in the same place, so they can understand why a circle is a kind of ellipse. You can also tell them that earth's orbit is an ellipse with one of the foci being the sun (but the earth's orbit is much closer to a circle than the ellipse you were drawing, unless your sundial is very far north or south--the sundial design becomes more circular the closer you are to the poles).

Step 4: Draw Hour Labels

You now need to mark tick marks on the ellipse for where the hour indicators are. How you do that depends on whether you drew the ellipse in Step 3.

Standard method (if you drew the ellipse)

The Step 4 page of the PDF file shows measurements for locating the tick marks for times from 4am to 9pm. Feel free to omit the earlier or late hours if they are useless to you at your location.

The PDF measurements show how far each tick mark's intersection with the ellipse is from the nearest tip of an axis. The PDF file shows blue double-headed arrows between the axis tip and the hour tick mark, and in the middle of each blue double-headed arrow there is a measurement. The distances are straight-line distances. The PDF file gets crowded at this point--I couldn't figure out how to lay it out better (remember that every location gets a somewhat different diagram so it would need to be automated)--but you should be able to figure out which measurement goes with which hour. Obviously, the smallest measurement in each part of the dial goes with the hour point closest to the axis tip.

I used a yardstick to measure out these distances to about 1/4" precision. (Also, because my east-west "horizontal" line was a gap between paving stones, I made sure to measure consistently from the side of the paving stones where I stuck the pins.) I then marked the times. I wanted to use Roman, but my six-year-old insisted on Arabic, so we went with that.

You can further subdivide each hour into quarters if you like.

Ellipse method without drawing (if you can insert pins but can't draw the ellipse)

This is the method to use if you're working on grass or some other surface where you can't draw, but where you can still attach pins (screwdrivers, rods, pencils). Mark the center and the endpoints of the two axes with stakes (sticks, dowels, bits of bamboo, etc.--I'll just call them "stakes"). Measure the locations of the pins (screwdrivers, thin dowels, sticks, etc.) at the ellipse foci, as in Step 3, and insert the pins (or have someone hold them in place).

Now, put the loop of string around the pins, and refer to the Step 4 page of your PDF printout. The step will give you distances between the hour points and the nearest axis ends. Put a thin stick (pencil, screwdriver, finger, etc.) in the loop of string (remember the string goes around two pins), and go approximately to the location of the hour point you want to mark. Keeping the loop taut, measure off the distance between your stick and the axis end, as marked on the Step 4 page. Affix the stick (or stake or just place a distinctive rock) in the place, and repeat with a new stick for the remaining hour points.

When done, you have all the hour points marked. Then you can put in stones with times painted on them, stakes with hours written on them, sticks with flags with hours written on them, flagstones with embossed hours, etc.

(x,y)-coordinate method (if you couldn't draw the ellipse and can't attach pins)

Drawing the ellipse is lots of fun. But if your surface doesn't allow it, or you can't affix pins in any way, you need to run my sundial generator in (x,y)-coordinate mode. You will then have a Step 4a and a Step 4b page. On the Step 4a page, under each hour you will have its (x,y) coordinates. You then use the x coordinate to measure off the distance to the left or right of the vertical line (directions when holding the PDF page with the text showing upright), where positive numbers are to the right and negative to the left, and the y coordinate to measure off the distance up (positive) or down (negative) from the horizontal line. For instance, for my location and dimensions, using feet and inches as my measurement, 1 pm is marked with the coordinates (-(0'8 7/8"),3'1 1/4"). The means I measure 8 7/8 inches to the left of the vertical line, and 3 feet 1 1/4 inches up from the horizontal line, and put a dot there. When the shadow of the gnomon will be centered on that dot, it's around 1 pm. You can put a paving stone there or stick a flag in the ground or mark the place in some other way.

You can then optionally use the Step 4b page to verify that you placed everything in the right place by now measuring the same distances that were used in the straight-line method.

Step 5: Draw Monthly Gnomon Position Tickmarks

This step and the next are what make this an analemmatic sundial. If you had a fixed pole in the ground, and every day at the same time (correcting for daylight savings if needed) you drew on the ground where the shadow of the tip of the pole appears, that shadow would trace a figure-eight curve on the ground called an analemma (due to the tilt of the earth's poles, the ellipticity of the earth's orbit and the tilt of the earth's orbit). But we want our sundial to show the same time each day at the same time! The analemmatic sundial solves this problem by moving the gnomon--the vertical item that casts the shadow--to different positions at different times of year to correct for the sun's position going around the analemma. (Actually, this only corrects for the variation in declination. There is a correction table for the other--smaller--kind of variation in the next step.)

In this step, you will draw some tick marks on the vertical axis as the start of the adjustment locations. These tick marks are parallel to the horizontal (east-west) line, and the Step 5 page of your PDF file shows for each tick mark how far it should be put above or below the horizontal line. The diagram also shows the direction in which they stick out (the ones at the top and bottom stick out both ways).

If you're using a surface that doesn't let you draw, put sticks on the ground or otherwise mark the locations.

Step 6: Finishing the Sundial

Now extend the tick marks a little, and between them write the names of the months as on Step 6 of your PDF file. The spaces for January and December will be rather tight--if you can't fit them, then draw an arrow from your label to the spaces, just as in the PDF file.

Step 6 of your PDF file will also carry a summer/winter time reminder (depending on what daylight savings option you chose). If you like, write it down somewhere on your sundial. For instance, my sundial is set to summer time right now, and has the reminder "Subtract an hour for winter time." In other words, if it's winter time (i.e., standard, non-daylight-savings time), and the clock shows "1:30 pm", it's actually 12:30 pm. Of course, since mine is made with chalk, I can just relabel it when the time changes.

There is also a correction table for the sundial based on the Equation of Time. This gives the number of minutes to add or subtract to the clock reading depending on the date. For instance, around August 15, you should add four minutes according to the table (this table is the one thing in the design that doesn't depend on where on earth you are!). The maximum correction is fifteen minutes, so if you don't mind your clock being off by a maximum of 15 minutes (it's a sundial after all, not a precision Swiss watch!), you can skip it. Or you can paint it on, or include a plaque with it.

At this point, you should try out your sundial (see the next step of the Instructable for usage instructions). Once you are fully satisfied, you can go over it with sidewalk chalk (which is a bit harder to erase) or paint (which is much harder to erase). I suggest you test it out at a couple of times of day.

Step 7: Using the Sundial

To use the sundial, the gnomon should be placed vertically upright on the central (north-south) axis of the sundial, as close as possible to the correct date. For instance, if it's August 1 or July 31, you put it right near the line dividing the July and August areas from each other. If it's August 10, you put it one third of the way into the August area, starting from the July area.

Notice that each position on the central (except the bottom and top ones) occurs twice a year. (That's because of the way the analemma loops.) For instance, the gnomon goes in the same place in late September as in mid March.

If the sundial is large enough that a person can act as the gnomon, just put your feet on either side of the north-south axis of the sundial, with your legs (not your feet--your feet stick out forward from your legs!) centered on the correct date. Then read the time in the center of where your body's shadow hits the ellipse. If the person is too short for their shadow to reach the ellipse at a given time, just imagine extending the line of their body until it reaches the ellipse or have them hold their hands joined together above their head.

If you want greater precision, hold something thinner than a person (yard stick, pipe, etc.) as precisely upright as you can over the correct date location on the central line, and also use the correction table (see Step 6). Depending on what material you're working with, you might be able to make holes for inserting a gnomon in different locations on different dates.

The second photo shows my shadow indicating a time around 8:45 am (the time-stamp on the photo reads 8:37--it's hard to get one's legs centered on the date). You will notice that there are also tree shadows. Unfortunately, our driveway has no spot that is consistently fully exposed--there are a lot of trees. However, the photo also shows that it's quite possible to read the sundial even if there are shadows of foliage. That's in part because the shadows from trees are much less sharply focused than shadows cast by the person who is much closer.

The third photo shows my shadow indicating a time around 11:30 am. The photo was taken at 11:37 am. Notice how in the morning my shadow extends far beyond the edge of the sundial, but not so closer to noon, despite the fact that the sundial is elliptical. For the record, I am six feet tall (about 180 cm).

The last photo shows my daughter's shadow at 4:30 pm. She is about 4.5 feet tall (about 135cm). The sundial would probably be somewhat too large for her around 1 pm--one would have to extrapolate from the shadow.

There are, as I said, many variations on this theme. I'd love to see links in comments to photos if you've made such a sundial.