A Hemispherical Paper Sundial

If you watch the sun cross the sky day after day, what do you notice? Viewing the sky above you as a dome, the sun rises every day on the eastern side of the dome, angles upward into the southern sky (assuming you live in the northern hemisphere), reaches a peak at noon, and then descends down the western wall of the dome, finally crossing the horizon in the west at sunset along another diagonal. In appearance, the sun circles around us each day as if on a tilted wheel. And each year, this daily circle gradually oscillates between northern and southern limits. In summer, the sun climbs higher in the sky and exposes more of the circle, giving us more of its light and warmth. In winter, it stays lower, keeping more of the circle hidden, and giving us less of its light and warmth.

If we were to draw these paths on a real dome, it might look something like a tilted "highway" arcing across the southern side of the dome, with the solstice paths resembling the shoulders of the highway, and the equinox path resembling the centerline.

What if we made a real dome, with the paths traced upon it, but we turned the dome upside down and backward, so that it became a bowl with the "highway" running across the northern side? What if we then placed something in the center of the bowl to cast a shadow? The shadow would fall on the "highway" on the bowl in exactly the opposite position to the sun's position in the real "highway" in the sky overhead. With suitable markings, we could track the sun's progress, not only through the hours of the day but also through the seasons of the year. We would have a working replica of the sky overhead.

The first scientific instruments in history were probably sun-tracking sundials like this, made by the Ancient Greeks. They carved hemispheres (among other shapes) from stone, and placed metal needles in the center as the pointers (or "gnomons") for their dials. Sadly, if you try to find Ancient Greek sundials 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. Greek sundials are over two millennia old. Korean astronomers made a few hemispherical sundials 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 first attached picture shows a hemispherical sundial made out of paper, in sunshine. The sideways curves across the north side of the bowl mark the path of the gnomon's shadow in various months. The highest curve (corresponding to the lowest arc of the sun across the sky) marks the path on the winter solstice, and the lowest curve (corresponding to the highest arc of the sun across the sky) marks the path on the summer solstice. The intermediate curves mark one-month intervals. The north-south lines mark equal intervals of one hour, or one twenty-fourth of the circle, with the central north-south seam corresponding to noon. The picture shows the sundial at roughly 3:30 in the afternoon, about one month before the winter solstice.

Like any sundial, a hemispherical sundial needs to be custom-designed for your latitude. How far you are from the Equator determines how "tilted" the sun's circle is in your sky, and any sun-tracking sundial needs to reflect this. I have provided templates for hemispherical sundials in ten-degree increments of latitude. Pick the latitude closest to your own and download the corresponding template.

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.)

I have designed two possible gnomons: a "protractor gnomon" and an "arrowhead gnomon". The latter 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 around the outline. The first attached picture shows what the two gnomons look like after they have been cut out and folded, but not glued.

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 gray triangular wedges, leaving a right-angle corner at the center of the semicircle. This corner will function as the shadow-casting "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, making a sort of circular accordion. 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 of the four flanges, 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.

The second attached image shows what the two gnomons should look like after gluing.

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. There is a dot, surrounded by a circle, marking a special place in the sundial — the "sundial center." This is where the dagger gnomon will be mounted. The overlap area is missing from the seam containing the sundial center, so that the gnomon can be mounted along this seam.

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 ends of two adjacent brackets.

Once all of the petals have been glued in place to all of the brackets, forming the bowl, 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, so "North" on the trim ring should be aligned with the central seam on the highway side of the bowl. (If you live in the Southern Hemisphere, the "solar highway" should lie on the southern side of the bowl, and if you have chosen the design for the Equator, the bowl will be symmetric, so either way will work. The attached picture shows a Northern Hemisphere 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. In the bowl of the sundial, there will be a black dot at a special location along this seam: the "sundial center." The "protractor gnomon" is decorated with an "axis" or "axle", and this axis on the gnomon must be aligned with the dot on the bowl. These markings symbolize the pole and the axle of the "wheel" 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 these markings symbolize the axis of the Earth itself. In other words, the protractor gnomon is decorated with an axle that will be parallel to the Earth's axis in reality, and that will point towards the North Star in the sky (if you live in the Northern Hemisphere). The markings for the solstices and 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 "dial center" 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 will make very little difference if you put the gnomon in upside down. (And if you do live near the equator, it should be fairly obvious which way the gnomon needs to be inserted.) Finally, lock the gnomon into place by bending the flaps sideways.

When inserted, the gnomon is meant to be centered over the dot marking the "sundial center", and should fill the surrounding black circle on the bowl. Esthetically speaking, I think this is definitely the prettiest way to place it. And if you want the axis of the gnomon to show you the alignment of the Earth's axis, then this is where you need to place it. But practically speaking, it isn't really crucial that you mount the gnomon in the correct place along the seam. It could be higher or lower and still work as a sundial. 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.

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 instead, or a simple pair of crossed supports. 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.

Start by selecting the design you want, downloading the template, and printing the design onto card stock. I think the two "simple" stands are easy enough to make without instructions. If you want to make the "fancy" stand, proceed by following the instructions below.

The fancy stand is composed of four support struts at the corners, and four folded braces joining the struts.

To make the four folded braces, score all of the dotted fold lines, cut along all of the black lines, including the incisions alongside the gray glue areas, and then fold all of the dotted lines as valley folds. Then glue each gray glue area to the neighboring triangle. One of the attached pictures shows the braces in various stages of assembly.

The four support struts also need to be scored along the dotted fold lines, cut out along the black lines, folded (all as ridge folds this time), and glued together. I haven't included any gray "glue areas" on the parts to indicate where to glue, 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, a glued support strut should have a rectangular top and bottom, with two triangles protruding from one side. One of the attached pictures shows the support struts in various stages of assembly.

Finally, glue the triangular ends of the folded braces to the insides of the triangles of the support struts to make a structure resembling the one in the photograph.

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 of the stand. 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, which will place the gnomon halfway in between the SW and SE struts.