Naples: A Disguised Analemmatic Dial

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aples: A Disguised Analemmatic Dial A Gnomonic xplanation by Frank H. King The ew Dial nearing completion In the autumn of 200 a novel analemmatic sundial was laid out in aples.

1 aples: A Disguised Analemmatic Dial A Gnomonic xplanation by Frank H. King The ew Dial nearing completion In the autumn of 200 a novel analemmatic sundial was laid out in aples. The hour points are arranged in a circle instead of in an ellipse. As the diallist who undertook the calculations, I have written an explanation which is presented as a sequence of diagrams on the following pages. Many people were involved in this project including architects, designers, stone-cutters and surveyors. A proper list of credits will be added later. At present, this explanation is concerned with answering a single question: o Circle, o Dial How can this sundial work if it breaks the usual gnomonic rules? The design studio that asked me to undertake the calculations provided a straightforward specification: 1. e want a very simple human sundial [meaning an analemmatic sundial with a vertical gnomon] for the latitude of aples. 2. To keep the sundial simple, there should be only three points on the date line: one for the summer solstice, one for the two equinoxes and one for the winter solstice. 3. The finished sundial must be in the form of a circle m in diameter. 4. The hour points must lie in a circle whose diameter is just under m. The first three requirements are easy to satisfy. The final requirement is impossible to satisfy but it came with a clear message... o circle, no dial. How could I achieve the impossible? Read on...

2 1 A Conventional Analemmatic Dial Fig. 1 A Conventional Analemmatic Dial for the Latitude of aples The latitude of the site is 40 1 and a conventional analemmatic sundial which satisfies the first three requirements is shown in Fig. 1. The hour points lie on an ellipse and the date line lies on the minor axis of the ellipse. The date line is aligned north south. Point shows where to stand at the summer solstice. Point shows where to stand at the equinoxes and point shows where to stand at the winter solstice. This is the diagram that I would normally give to a design studio. I would provide the coordinates of each hour point and of each of the points on the date line. These would be expressed in terms of an X Y system of coordinates in which the origin is the centre of the ellipse, the X-axis runs west east along the major axis, and the Y -axis runs south north along the minor axis. As a check, I would also provide the radial distance of each point from the centre. The design team in the studio could then embellish this outline in any way they wished. The hour points in Fig. 1 do not lie in a circle so this design does not satisfy the fourth requirement. evertheless, Fig. 1 serves as a reference analemmatic design which will be referred to in what follows.

3 2 Introducing Circles Fig. 2 The Analemmatic Dial set out in a Circular Ring of tone Fig. 2 shows a grey circle drawn with a very broad brush. The brush is so broad that it accommodates the hour-point ellipse. uch a circle could be formed from stone slabs and provides a naïve way to satisfy the fourth requirement. The hour points undoubtedly lie in a circle but not quite in the way that the design studio had in mind!

4 3 Distorting the Analemmatic Dial Fig. 3 A Distorted Dial The design studio noted that by stretching the dial in Fig. 1 in the north south direction but leaving the west east dimensions unchanged, the ellipse could be distorted into a circle. The result is shown in Fig. 3. This is what the design studio really wanted but, alas, it works satisfactorily only at the north pole. This false start will be referred to later (see Fig. ) when it will be shown not only why it fails but also how it can be used as a starting point for a successful design.

5 4 An llipse Constructed from Circular Arcs Fig. 4 Four Circular Arcs Fig. 4 illustrates a traditional way of drawing an approximation to an ellipse using four well-chosen circular arcs. The centres of these arcs are indicated by crosses. Although Fig. 4 employs four circles rather than just one circle as required, the construction hints at a solution... The arc on the north side passes almost exactly through all the hour points from to inclusive. Can one draw a slightly different circle that, at least approximately, incorporates the early morning and late afternoon hour points too?

6 Two Trial Circles Fig. Circles through some of the Hour Points A circle, centred on the north-south axis, can be drawn through any pair of morning hour points and will, by symmetry, pass through the matching pair of afternoon hour points. Two such circles are shown in Fig.. One circle passes exactly through the hour points,, and and the other passes exactly through the hour points,, and.

7 A Circular Analemmatic Dial Fig. A Circular Analemmatic Dial with just four hour points Fig. shows the larger of the two circles of Fig. with just the hour points,, and. The date line and the hour points,, and are exactly as in the conventional analemmatic sundial of Fig. 1. Can other hour points be added? Before answering that question, it is worth considering a quite different procedure for producing the outline design shown in Fig..

8 An Alternative Procedure Fig. Four Lines and five Points Fig. illustrates why the distorted dial in Fig. 3 fails. Point marks the centre of the circle. The line drawn from shows the direction of the shadow of a vertical gnomon at at h at an equinox. This intersects the circle at hour point. The point on the centre-line is chosen so that the shadow of a gnomon at at h at the summer solstice also intersects the circle at hour point. A second line through shows the direction of the shadow at h. This intersects the circle at hour point. The point on the centre-line is chosen so that the shadow of a gnomon at at h at an equinox also intersects the circle at hour point. Points and do not coincide. One cannot have two equinoctial points (one for h and one for h) so the design fails.

9 Developing the Alternative Procedure 12, Fig. Four Lines and four Points The approach used to construct Fig. can be made to work. If the starting point is moved northwards, the gap between and becomes smaller and a position can be found where and coincide. This is shown in Fig.. In addition, the point on the centre-line is chosen so that the shadow of a gnomon at at h at the winter solstice intersects the circle at hour point. Fig. is simply a rediscovery of the date-line points, and and the hour points and in Fig.. The two approaches arrive at the same conclusion. ince the sun is always due south at noon, hour point 12 can be added where the centre-line and the circle intersect. This, though, is slightly to the north of hour point 12 in Fig..

10 Filling in the other Hour Points Fig. Three ets of Hour Points Using the three points, and of Fig., Fig. shows three sets of hour points. Those marked with an open circle are appropriate at the summer solstice for a vertical gnomon at. Those marked with a solid circle are appropriate at the winter solstice for a gnomon at. Those marked with a cross are appropriate at the equinoxes for a gnomon at. At,, 12, and there is exact agreement. At, 11, 13 and the agreement is good and one can simply average the positions in each triplet. At,,,, 1 and the agreement is poor but, at these hours, the sun is either below the horizon or very low above the horizon except in summer. Accordingly, the points appropriate for the summer solstice (the open circles) can be used.

11 The Completed Outline Design Fig. The best-fit Hour Points Fig. shows the outline that was handed over to the design studio to develop into a full-size paper sundial that could be laid out on site. As noted on the previous page, hour points,, 12, and are correct for each of the date-line points, and. ach of the hour points, 11, 13 and is the best-fit approximation to three distinct points. Hour points,,,, 1 and are correct for date-line point only.

12 11 Augmented Outline Design Fig. 11 The best-fit Hour Points and the Original llipse Fig. 11 shows the previous figure augmented by the ellipse which runs through the hour points of the reference analemmatic sundial. For the hours to, the ellipse runs through the new hour points exactly or very closely. Hour points,,,, 1 and are correct for a gnomon at at the summer solstice but not at other times of year when, at these times of day, the sun is either below the horizon or so low that, at the site, the sundial is in shadow. ote (for example) that a line drawn from date-line point to hour point intersects the ellipse at the position of hour point on the true analemmatic dial shown in Fig. 1. The point of intersection is marked by an extra dot.

13 12 An Analemmatic Dial on loping Ground Fig. 12 A loping Dial Overlaid on the Original hen the paper dial based on Fig. was laid out on site, it was noticed that the ground slopes quite steeply. The line of greatest slope (the broken line in Fig. 12) makes an angle of 1 42 to true north south. This line is inclined (upwards to the left) by an angle of to the horizontal. The solid red line through the centre is at right-angles to the line of greatest slope and, assuming that (locally) the ground is a plane, this line is horizontal. etting out a conventional analemmatic sundial on sloping ground is not difficult. The original analemmatic design of Fig. 1 is projected orthogonally onto the sloping ground. In practice the slope is so small that the adapted sundial is almost indistinguishable from the original. Fig. 12 shows the revised dial, in red, rotated about the horizontal so that it lies flat on the original design. The original ellipse is stretched very slightly in the direction of the broken line but the dimensions in the direction of the solid line are left unchanged. It is just possible to see that the red dot for hour point is not quite centred on the black point for this hour.

14 13 xaggerating the lope Fig. 13 xaggerating the slope Fig. 13 is identical to Fig. 12 except that the slope has been increased by a factor of four to emphasise the stretching in the direction of the line of greatest slope. ote that the date line appears no longer to be aligned north south. This is because the sloping, red, dial has been rotated so that it lies in a horizontal plane. hen the red dial is set out on the sloping ground the red dot labelled is vertically above the corresponding black dot. The date line is in a vertical north south plane but it is not horizontal. Unfortunately, an orthogonal projection of the circle used in Figs to 11 onto the slope changes the circle to an ellipse. The fourth requirement would no longer be satisfied. It was necessary to make a fresh start. The procedure described for Figs and was used again but the circle was assumed to be on sloping ground rather than level ground.

15 The Hour Points on the loping Circle Fig. Revised Triplets of Hour Points Fig. corresponds to (and is almost indistinguishable from) Fig.. The main difference is that the three date points, and have been determined for a circle that is to be set out on the sloping surface of the intended site. Three sets of hour points are again shown and these are appropriate for a vertical gnomon at, and respectively. As before, an averaging process is used to chose hour points for the hours h to h and, for the early morning and late afternoon hours, the points appropriate for the summer solstice (the open circles) are used.

16 The Final Outline Design Fig. Revised best-fit Hour Points Fig. shows the final version that was handed over to the design studio. This was used as the basis for the sundial that was actually made and which is now in position in aples. If the sun is shining on the site, this dial will correctly indicate local solar time. Although it is not a true analemmatic sundial, it should be used exactly as though it were such a dial. Churchill College Cambridge, CB3 0D

17 Trying out the Paper undial Inscription by Virgil: All our sweeter hours fly more quickly Italian urveyor and nglish Diallist

18 The Carrara Marble Quarry Lifting out Blocks of Marble Cutting Marble using a ater-jet Cutter

19 Brass Lettering Brass Cut-Outs Brass Cut-Outs Inlaid into the Marble

20 etting out the Marble Blocks on ite The Central tone and the Date Line All tone Blocks in Position

Polar Sundial. Cut carefully on this line. Cut on this line.

Polar Sundial. Cut carefully on this line. Cut on this line. 1 The Sundial Primer - "Dialling Guides" The purpose of the "Dialling Guides" is to provide an easy method for laying out the hour lines for a number of polar sundials located at any latitude in either