Hemispherical Sundials

Paper models of ancient Greek sundials, with solstice and equinox curves, and an optional map. (This is one of two paper models of Ancient Greek sundials. I made this youTube video to introduce the pair of them.)

I'm fascinated by sundials. I especially admire the classic sun-tracking instruments invented by the Ancient Greeks. These masters of geometry and of stone sculpture managed to freeze the graceful geometry of the sky, and etch it permanently into beautifully carved cones, cylinders, and hemispheres, as well as onto flat surfaces.

These forgotten instruments should be brought back into science education. These simple sun-tracking devices help you to see before your very eyes the relationship between the ground under your feet and the sun in the sky over your head. They permanently capture in stone (or paper) the "highway" of the sun across the sky, and with a pointing device or "gnomon" they allow you to use this track like an enumerated gauge or dial, and to measure the position of the sun in its course across the sky at any given moment.

The best for this is the hemispherical sundial, which is essentially an inverted model of the dome of the sky. This paper sundial is my own updated version of a classic Greek hemispherical sundial.

The Sun’s “Highway” Across the Sky

Science begins with careful observation. The science of astronomy begins with methodical observation and record-keeping of what goes on in the sky. My favorite way to begin astronomy with children was to say: "Wouldn't it be fun if we could reach up with a giant marker, and draw on the sky, and record where everything is and where it goes?" Thinking of alternative ways to do this, and then implementing some of them, was a fun and instructive exercise. One way would be just to put an empty bowl in the the sunshine, with something in the center to cast a shadow, and then we could draw shadows on the bowl instead of drawing the sun on the sky. Another alternative might be time-lapse photography.

Does the sun stop and go in the sky? Does it turn corners or swirl around in zig-zags? Of course not. If we study the motion of the sun, we find that it just passes across the sky in steady arcs, like portions of a circle around us, with the rest of the circle hidden below the horizon. It looks as if the sun is turning on a wheel and going in circles around us. But the wheel is tilted, for some reason. The sun rises and sets at an angle, and it never actually goes straight overhead (assuming we live in the Northern Hemisphere). If we study the sun for many months, we can also discover that its arc is higher in the summer, and lower in the winter. More than half of the circle is above the horizon in summer, and less than half in winter, which is why we have more than 12 hours of daylight in summer, and less than 12 hours of daylight in winter.

Altogether, if we could draw the sun's "highway" across the sky for a full year (and assuming that we live in the Northern Hemisphere), we would see something like this:

The "Highway" of the Sun Across the Sky

Every day, the sun rises on the eastern side of the "dome", angles upward into the southern sky, reaches a peak at noon (but does NOT go straight overhead), and then descends down the western wall, eventually crossing the horizon in the west along another diagonal. And this daily circle gradually oscillates once a year between a northerly and southerly limit, giving us summer and winter.

So to summarize: It looks as if the sun is turning around us on a wheel, except the axle is tilted up into the northern sky for some reason, and the wheel slides up-and-down, back-and-forth on the axle, once a year. The daily turning of the wheel gives us day and night, and the annual oscillation of the wheel between the northerly and southerly extremes gives us summer and winter.

All of the rhythms of life on earth are driven by these cycles of the sun. The primary concepts of "day" and "year" and all of our tools for measuring the progression of time are rooted in measurement of these cycles of the sun. The modern clock and the modern calendar both evolved from simple markers for solar events. Once each day there is a special moment when the sun is at its highest point, shadows are the shortest, and the sunlight is half over for the day. In English, we call this moment "noon." The opposite is "midnight," and to this day "noon" and "midnight" are the checkpoints of our daily clock. They are to the clock what cardinal directions are to a compass. Once each summer and once each winter, the circle of the sun reaches its northerly or southerly limit for the year, stops, turns around, and heads back in the other direction. We name these two special "turn-around days" the "solstices". Once each spring and once each fall, the circle passes through the midpoint, on which days the sun rises due east and sets due west, and the circle is divided into two equal portions of 12 hours each. We name these special days the "equinoxes." (The word "sol-stice" comes from Latin words meaning "the sun's standstill", or "the station of the sun." The word "equi-nox" comes from Latin words meaning "equal night.") And these four annual events are the checkpoints of the calendar. Or at least they are with regard to astronomy and to meteorology. The modern 12-month civil calendar is no longer lined up with the four annual solstice and equinox events, but it was when it was invented. The calendar was originally designed so that each solar event would mark the beginning of a new month. It might have been a little like having New Year's Eve and New Year's Day four times a year. In the modern world, all solstices and equinoxes occur about 3/4 of the way through their respective calendar months, and nobody notices when they pass.

So to be geometric, we can mark the solar "highway" across the sky with a "centerline" marking the sun's path on the equinoxes, and two "shoulders" that mark the sun's path on the equinoxes, as well as the hourly marks measuring the sun's progress along the highway. This is what I've depicted in the sketch above.

An Inverted Sky

Now, what happens if we make a physical model of the sky, complete with clock and calendar marks drawn upon it, and we turn it upside down and backwards? What if we put something in the center to cast a shadow? We have made a way to capture in writing the "highway" of the sun across the sky, and we can use it to track the real sun. The "highway" on the bowl becomes like the scale of a dial, and the pointer shows by its shadow on the scale the position of the sun in the sky. We have made a "sun-o-meter." Here's what a sunometer looks like in late afternoon (in the Northern Hemisphere), about a month before the winter solstice:

An Upside-Down Model of the Sky

At any moment, the sunray passing across the tip of the shadow-caster will land on the bowl at a position that mirrors the position of the sun in the sky. When the sun is high in the sky, the shadow will be deep in the bowl. When the sun is low in the southwestern sky, the shadow will fall high on the northeastern wall of the bowl. And so on. The bowl is a working replica of the sky.

The Ancient Greeks made sundials like this, carved from stone, with metal needles as the pointers for their dial. Sadly, if you try to find them in a museum, or in a Google search, you won't find many examples, and the ones you find will probably be in ruins, with missing gnomons. The Greek sundials are over two millennia old. Korean astronomers also made a few out of bronze or iron less than one millennium ago, and several of these — I'm not sure how many — still survive in good condition. There are also a couple of hemispherical sundials at Jantar Mantar in India, built a few hundred years ago.

You can also make your own out of paper.

The Half-Globe Sundial

When I first had the idea to put maps on sundials, I thought: "This is going to be so awesome … !" So I put maps on these hemispherical sundials, and also on my Butterfly Sundials. I like the way they turned out on the Butterfly Sundials, but on these hemispherical sundials, I think they might almost be too distracting. Especially for educational purposes, I think I like the plain black-and-white sundial versions better. If I were still teaching in a classroom, I definitely wouldn't begin with a sundial with a map on it. But they are still pretty cool, if I do say so myself, and they could make a fun additional project after making a basic sundial. So here's an example of a hemispherical sundial with a map on it:

A Half-Globe Sundial (Designed for Des Moines, Iowa)

What's the point of having a map on a sundial? Notice the special way in which the map has been positioned on the sundial above. A half-globe has been drawn on the inside of the bowl, but the user's location (in this case, Des Moines, Iowa) has been centered at the bottom, and the cardinal directions have been reversed, to match the shadows falling opposite to the sun. Notice how this makes the globe line up with the solar "highway" on the sundial. The summer solstice curve of the sundial now lines up with the Tropic of Cancer on the map, the winter solstice curve lines up with the Tropic of Capricorn on the map, and the equinox curve has become the Equator. The "Solar Highway" of the sundial, i.e. the area where the shadow can land, lines up with the "Tropics" or "Solar Belt" of the Globe, i.e. the area of the Earth where the sun can be directly overhead. And best of all, wherever the shadow lands on the map is where the sun is directly overhead at that moment.

With a properly-aligned map on a sundial, you can not only track the progress of the sun across your sky, you can also track the progress of the sun around the world.

Throughout my teaching career, I have wanted above all to avoid "floating abstractions." Science is so much more fascinating when you can weave a web of strong connections between the ideas in your mind and the personal evidence of your eyes and your ears. Ideas that have no such connections are just fantasy. In astronomy, sundials can help to provide such links. The sun-tracking sundials like the ones made by the ancient Greeks recorded in visible form the geometry of the sky, and were probably the first scientific instruments in history. For children learning astronomy, tracking the sun and making simple sundials should be among the first lessons. And maybe after they learn all about solstices and equinoxes and the latitudes and longitudes of the globe, they can make a sundial with a map on it and track the sun around the world.

A Sundial For Your Location

Before we get to the downloads, I should probably warn you that these paper hemispherical sundials are a pain in the neck to make. Expect each one to take a couple of hours, and maybe longer if you want to make the fancy stand as well. If you want an alternative sundial with similar content that you can make easily in five minutes, I suggest the Butterfly Sundials. Those contain much the same information, and they are very easy to make. I'm quite proud of those.

But if you like a challenge, and you want to make something that nobody else in your neighborhood has, then by all means please help yourself to one of the following downloads.

Black-and-White Sundials

As with my other sundials on this website, I'll offer a list of free downloads, and you will need to choose the design for the latitude closest to your own. However, these will be plain black-and-white sundials, with no maps.

If you study the sun's path across the sky in different parts of the world, you can see that the tilt of the sun's "wheel" in your sky depends on how far north or south of the equator you live. (If you are a teacher, perhaps my slideshow on the circles of the sun could help you to establish this point.) This means that every sundial must be custom-designed for the latitude in which it is to be used. In the menu below, there are links to PDF files containing hemispherical sundials designed for every 10 degrees of latitude.

Custom Half-Globe Sundials

If you want a customized hemispherical sundial with a color map on it, designed for your exact latitude and longitude, you can use the form below to create printouts for the parts. I intend to charge a fee for this service eventually, especially if I manage to improve the design. But for now I'm making it available to anyone with the access code, and I'm giving away the code. For the near future, it's "Hipparchus". A donation is not necessary to use the form (for now), but I would appreciate enormously any support for my efforts to make science education more fascinating and more active-minded.

Custom Color Sundial

A Half-Globe Sundial with a color map, designed for your exact latitude and longitude. Note that latitudes are considered negative in the Southern Hemisphere, and longitudes are considered negative in the Western Hemisphere. If you live in the New World, your longitude is negative. Don't forget to SAVE the file after you receive it!

Access Options

If you click on the "Use Current Location" button above, it should automatically fill in your current GPS coordinates, but you may need to enable location services in your web browser. Also note that the accuracy is usually very good, but can occasionally be off by several hundred miles depending on which method your device uses to calculate your location. You can also probably find your current coordinates in the "Compass App" on your mobile device, and you can find the coordinates for any location on Earth by right-clicking on any location in Google Maps.

In any case, as long as you are within a degree or so of the location you want, you probably won't notice any difference in the final design, unless you examine it closely with a magnifying glass.

Making the Sundial

The physical design of the color Half-Globe Sundial is identical to that of the black-and-white Hemispherical Sundial, it just has different markings. So the instructions are identical for making them both, with one or two exceptions. If you have the color version, and if you can access the "layers" tool of your PDF reader, you have a couple of printing options. You can deselect the "map" layer to omit the map and make a simple black-and-white sundial for your exact latitude. You may want to do this to use for a simplified educational lesson, or to make a practice sundial before you use up all of the color toner in your printer. You can also turn off all of the markings, if you want to make a blank paper hemisphere for any reason.

Please note that if you have the color version, and you turn off the map layer, the markings on the "protractor gnomon" will also change. The "protractor gnomon" has map-related markings on one side, and sky-related markings on the other side. But if you turn off the map on the bowl, all of the map-related markings will be replaced by sky-related markings, making both sides of the gnomon symmetric.

Whether you have the black-and-white or color version, and regardless of latitude, there will be three pages. The first page contains a flower-shaped pattern that will form the bowl, the second page contains brackets and a trim ring that will form the upper rim of the bowl, and the third page contains two alternative options for the shadow-casting "gnomon" of the sundial. Start by printing all three pages onto medium-weight card stock. I've never actually tried making a hemispherical sundial out of normal typing paper, because I'm pretty sure that such a thing would be impossibly limp and floppy. Heavy-weight card stock would probably work, but I would worry that the sides of the bowl might not curve easily enough, and the bowl might end up looking more like a pyramid than a hemisphere.

Making the Gnomons

I have designed two possible gnomons: an "arrowhead gnomon" and a "protractor gnomon". The former is a little more decorative, and is intended to be the more permanent piece, but the removable "protractor gnomon" is labeled with lots of information. They are interchangeable, but you can glue the arrowhead gnomon in place if you wish.

Before you cut out either gnomon, use a straight edge and a pin or something to score all of the dotted fold lines. This will give much straighter, crisper folds. Then cut out both parts all the way around the outline. This is what the two gnomons look like after they have been cut out and folded, but not glued:

Folded Gnomons

To make the "protractor gnomon", fold it in half and glue the two sides together to form a semicircle. (Use the glue sparingly. If the paper becomes too wet, the resulting semicircle will be warped and not as attractive.) You may want to rest a heavy book on it while the glue dries to keep it smooth and flat. After gluing the sides of the semicircle together, cut out the triangles, leaving a right angle corner at the center of the semicircle. This corner will function as the "pointer" when you insert the semicircle into the bowl.

To make the "arrowhead gnomon", cut it out around the perimeter, and then make short incisions where indicated, to create foldable flaps extending sideways from the tabs on the base. (There will be four such incisions, unless you live near the equator or near the poles, in which case there will be two.) Now fold all of the fold lines. The radial lines should be folded alternately as ridge and valley folds. Each plain dashed line not ending on a tab should be a ridge fold, and each axis decoration ending on a tab needs to be a valley fold. Fold and glue the entire structure into a four-flanged triangular "dagger" or "arrowhead", with tabs at the bottom. When gluing each flange, I find it helpful to rest the flange against the edge of a table or book, and smooth the flange flat against the hard surface before the glue dries.

After gluing, the two gnomons should look something like this:

Assembled Gnomons

Hemispherical Bowl

Now for the fun part. The bowl will be formed from the flower-shaped design on the first page, and the polygonal brackets inside the circle on the second page. The circular ring will form a final decorative trim to be glued to the upper surface of the bowl. Note that each "petal" of the "flower" has a thin overlap area along the counter-clockwise edge. This will overlap the adjacent edge of the neighboring petal, and is simply to help hide the seams between petals in the assembled dome. This overlap area is missing from the seam containing the north pole (or the south pole, for southern hemisphere sundials), so that the gnomon can be mounted along this seam.

The Parts for the Bowl

The gray trapezoids at the end of each "petal" are tabs to be folded, wrapped around the brackets, and glued into place. (This may be the most annoying part of the assembly. Sorry about that.) Start by scoring the two fold lines along the base and across the middle of each trapezoid, and then cut out the flower shape all around the outline. Also cut out the ring and the brackets.

Now, working your way around the edge of the bowl, wrap each tab around a corresponding slot in one of the brackets, and glue it into place. Four of the petals will need to have their tabs wrapped around the half-slots on two adjacent brackets. (This is one of the first things I will change in any future designs. Again, I apologize.) This is what it should look like after gluing the first two adjacent petals to the first bracket, and then just before gluing all adjacent brackets together:

A Partially-Assembled bowl
A Partially-Assembled bowl

Once all of the petals have been glued in place to all of the brackets, the final step will be to glue the trim ring onto the top of the bowl. The "North" and "South" labels are meant to indicate directions in reality, in your yard or on your balcony or wherever you set your sundial. For the sundial to work, these must be aligned properly with the north-south seam of the bowl. If you live in the Northern Hemisphere, the "solar highway" marks should be on the northern side of your bowl, and if you live in the Southern Hemisphere, they should lie on the southern side.

If you have the color version with the map you might be tempted to glue the trim ring with the "North" label on the same side as the North Pole on the map. But remember that the bowl is inverted compared to reality. The "south" side of the map needs to lie on the "north" side of the bowl in reality, and vice versa. For the sun-tracker to work properly, the map needs to be backwards compared to the world around you.

Placing the Gnomons in the Bowl

The "protractor gnomon" is meant to be removable. You should be able to just rest it inside the bowl, along the north-south seam. On all sundials, there will be a black dot at a special location, the "sundial center". This marks the "axis" of the "wheel," i.e., the axis around which around which everything in the sky appears to rotate. If you are already convinced that it is actually the Earth that is spinning, and not outer space, then this axis is the axis of the Earth itself. The protractor gnomon is decorated with an "axle" or "axis" that should be parallel to the Earth's axis in reality. If you have the color version with the map, then one of the Earth's two poles should be visible on the map, and the axis on the gnomon should line up with that, too. The markings for the solstices/tropics on the gnomon should also align with the "solar highway" marks on the bowl.

The "arrowhead gnomon" can be glued into place if you wish, but I think it stays in place pretty well without glue. To fasten it in place, insert the tabs through the north-south seam in the dial, where the "circle center dot" is located. I've calculated the shape of the gnomon assuming that the single open flange of the gnomon faces downward towards the bottom of the bowl. (By "open flange", I mean the one formed where the two extremities came together, rather than the other three which were formed along a fold line.) However, unless you live near the Equator, it shouldn't make much difference which way you insert the gnomon. Finally, lock it into place by bending the flaps sideways.

When inserted, the gnomon is meant to be centered over the "sundial center", and should fill the surrounding black circle on the bowl. (In map versions, this circle should coincide with the Arctic or Antarctic Circle.) Esthetically speaking, I think this is definitely the prettiest way to place it. But practically speaking, it isn't really crucial that you mount the gnomon in the correct place. What is important is that the tip lies in the center of the hemisphere. You can check your alignment by gazing across the top of the bowl, from one end of any seam to the opposite end of the same seam, and ensuring that the tip of the gnomon is in line with the seam, and also level with the top of the bowl.

Making a Stand for the Bowl

I designed a fancy stand for the bowl of this hemispherical sundial, but it's almost as much of a pain to make as the bowl itself. If you just want a simple stand you can make in five minutes, you can make a simple conical frustum, or a simple pair of crossed supports:

Various Options for a Stand for the Hemispherical Sundial

One benefit of the simple conical stand is that you can rotate the bowl slightly if you need to, to help compensate if your table isn't perfectly horizontal, or if you have made a bowl for a latitude that isn't quite right. I think the two "simple" stands are easy enough to make without instructions. If you want to make the "fancy" stand, select it from the menu below, and then follow the instructions below the menu.

Stands for the Hemispherical Sundial

(Free download)

The fancy stand is composed of four support struts at the corners, and four folded braces in between. The braces or brackets should look like this after you cut them out, fold them, and glue the ends together:

The Folded Braces for the Stand

Similarly, the four support struts need to be cut out, folded, and glued together. I haven't included any gray "glue areas" on the parts, because it really doesn't matter which flaps are glued over which flaps. All flaps will overlap or be overlapped by another flap, but it really doesn't matter which way they go. In the end, there should be a rectangular top and bottom, with two triangles protruding from one side. In various stages of construction, the four struts should look like this:

The Corner Struts in Various Stages of Development

Now if you glue the ends of the folded braces to the triangles of the support struts, you should have something like this:

A Paper Stand for the Sundial Bowl

If you are using the "arrowhead gnomon", and you live within 30 or 40 degrees of the Equator, you may find that the tabs of the gnomon clash with the support columns. In this case, just turn the stand so that the four corners are aligned NW, NE, SE, and SW, instead of N, S, E, and W. You may prefer that orientation anyway.


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