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The mathematics of

Jill Vincent University of Melbourne

s early as 3500 ago, shadows of sticks were used as a primitive Ainstrument for indicating the passage of through the . The stick came to be called a “” or “one who knows.” Early Babylonian obelisks were designed to determine . The development of trigonometry by Greek mathematicians meant that lines could be determined arithmetically rather than by geometry, leading to more sophisticated sundials. In the first CE, the Roman architect and engineer described several types of sundials in his (Lennox-Boyd, 2006, p. 32), including hemispherical, conical and planar dials. Sundials are often constructed to commemorate special events, for example, the Bicentennial Park in Sydney. The many common designs of sundials, such as vertical, horizontal and analemmatic dials, can all be derived by projections of the basic equatorial dial (Lennox-Boyd, 2006). In this article I show how trigonometry can be used to calculate the positions of the hour lines for sundials, with a particular focus on the mathematics underlying a recently-constructed unique horizontal sundial at Piazza Italia in Melbourne. The words “vertical” and “horizontal” are used in their normal

sense, that is, in the direction of gravity and at right to this. Australian Senior Mathematics Journal 22 (1)

Equatorial sundials

Let us imagine the as a giant sundial (Figure 1). The Earth’s axis is tilted at an of 23.5° to the plane of its around the . As the Earth rotates on its axis, the shadow of a vertical stick at the pole would form a on the surface of the Earth parallel to the . If the circle is divided into 24 equal hour marks, the position of the shadow around the circle would give the time. Sundials based on this principle are called equatorial sundials. In any sundial, the part that casts the shadow is called the gnomon. For an equatorial sundial, the gnomon must be parallel to Figure 1. Representing the the Earth’s axis. In Figure 2, P is in the at lati- Earth as a giant sundial.

13 14 Australian Senior Mathematics Journal 22 (1) Vincent If plane asanellipse.Thesemi-majoraxis, For averticalsundial,thecircularequatorialdial isprojectedontoavertical Vertical sundial towards thepolestar). point tothesouthcelestialpole(innorthernhemisphere,itpoints perpendicular tothegnomon.Insouthernhemispheregnomonwill horizontal isequaltothelatitude.Thedialwithhourmarksmustbe tude sundial thereforeneedsafaceonbothsides. the Sunislowerinsky, theshadowfallsonunderside.Theequatorial falls ontheupperfaceofdial.Between22MarchandSeptember, when (22September)andtheautumnMarch),shadow directly inlinewiththegnomonsothereisnoshadow. Betweenthespring AtmidsummertheSunis hour linesareequallyspaced,with15°intervals. the gnomonmakingananglewithhorizontalequaltolatitude.The b Figure 3showsanequatorialsundialforthesouthernhemisphere,with is thesemi-minoraxisofellipse, L. Thediagramshowswhytheanglethatgnomonmakeswith Figure 2.Inclinationofthedialandgnomonanequatorialsundial. Figure hemisphere. 3.Equatorialsundialforthesouthern a, istheradiusofequatorialdial. The mathematics of sundials Australian Senior Mathematics Journal 22 (1) 15 dial. Figure 5. Side view of gnomon of vertical 5. Figure sundial. Figure 6. Relationship between the T of the Figure equatorial dial and the projected hour angle H of the verticalequatorial dial and the projected L = 30°, T. OBC, we and and is the projected H OAC in terms of L = 15°, at 10 am, T Figure 4. Projection of the equatorial dial to form the of the verticalProjection 4. dial. Figure is the hour angle measured T The gnomon of the vertical with the vertical (that is, an angle L with the vertical (that is, an angle in with the horizontal), as shown the side view in Figure 5. sundial makes an angle of 90°– Calculating the angle for the vertical sundial In the southern hemisphere, the vertical dial is -facing. Unlike the equatorial dial, the hour angles are not equally spaced. In Figure 6, angle from the north– line around the equatorial dial. For example, at 11 am, T and so on. Angle hour angle on the ellipse of the vertical dial. Using the equation for an ellipse and applying trigonome- try in can find H 16 Australian Senior Mathematics Journal 22 (1) Vincent 12 noon 10 am 11 am Time 6 am 7 am 8 am 9 am From Note that but: Equation fortheellipseis: noon linefor1pmto6pm. ( 37.7°)areshowninTable 1.Theanglesaresymmetricalaboutthe The hourangles,H,for6amto12noonaverticaldialinMelbourne ΔOBC Table 1 T (°) 90 75 60 45 30 15 0 H (°) 90.0 71.3 53.9 38.4 24.6 12.0 0.0 are theparametricequationsforellipse. Figure 7. Hour angles for vertical sundial. Figure 7.Houranglesforvertical The mathematics of sundials Australian Senior Mathematics Journal 22 (1) 17 is the semi-major b with the horizontal. The semi-minor L Figure 8. Vertical sundial on Box Hill Library, Victoria. Vertical 8. Figure Figure 9. Horizontal projection of the equatorial dial parallel to the Earth’s axis. 9. Horizontal projection Figure Figure 8 shows a vertical sundial on the almost north-facing wall of Box almost north-facing sundial on the shows a vertical Figure 8 Hill Library in Melbourne’s eastern suburbs. The shadow indicates the time indicates the The shadow eastern suburbs. Hill Library in Melbourne’s 10:20 am. is approximately (north–south) axis of the ellipse, For a horizontal sundial, the circular equatorial dial is projected onto a hori- For a horizontal sundial, the circular 9). As for the equatorial and vertical zontal plane as an ellipse (Figure angle sundials, the gnomon makes an Horizontal sundials , the radius of the equatorial dial. If () axis is a, the radius of the equatorial dial. 18 Australian Senior Mathematics Journal 22 (1) Vincent H been used.For11am,forexample, in Table 2.Thelatitude37.7°has tal sundialinMelbourneareshown In Figure10,angle Calculating thehouranglesforhorizontalsundial horizontal sundialforMelbourne. spacing ofthehourangles forthe hemisphere. Figure11 shows the horizontal sundialinthesouthern numbers areanticlockwisefora Unlike theverticalsundial, The houranglesaregivenby parametric equationsfortheellipsehorizontalsundialare of thehorizontaldial.Usingsameapproachasforverticalsundial, around theequatorialdial.Angle The hourangles,H = tan –1 (tan15°sin37.7°) Figure 10.Relationshipbetween thehourangleTofequatorialdial and theprojected hourangle H ofthehorizontaldial. T , forahorizon- is thehouranglemeasuredfromnorth–southline ≈ 9.3°. H is theprojectedhourangleonellipse Table 2.Houranglesforahorizontalsundialin 12 noon 11 am 10 am Time 9 am 8 am 7 am 6 am Melbourne T (°) 15 30 45 60 75 90 0 H (°) 19.4 31.4 46.6 66.3 90.0 0.0 9.3 The mathematics of sundials Australian Senior Mathematics Journal 22 (1) 19 (see Figure 12) at Piazza Italia in the inner (see Figure 12) at Piazza Italia Solaris Figure 12. Solaris at Piazza Italia, Carlton. Figure Figure 13. Relationship between height and position. Figure m from the centre of the sundial at which the person stands, m from the centre of the sundial Figure 11. Spacing of hour lines for a horizontal sundial in Melbourne. 11. Spacing of hour lines Figure Garden sundials are typically horizontal dials. A large-scale horizontal are typically horizontal dials. Garden sundials If we consider a right-angled formed by the gnomon and If we consider a right-angled triangle sundial has been constructed in the New South Wales town of Singleton. the New South Wales sundial has been constructed in Horizontal sundial at Piazza Italia Horizontal sundial at Piazza The horizontal sundial the distance d an angle of 37.8028° (the local latitude) the hypotenuse of this triangle makes with the horizontal (see Figure 13). Melbourne suburb of Carlton has a unique design. Instead of a gnomon set Melbourne suburb of Carlton has stands on the north–south line at a calcu- at the angle of the latitude, a person height and acts as a vertical gnomon. lated position according to their 20 Australian Senior Mathematics Journal 22 (1) Vincent iue15. Figure The tipoftheperson’s shadowwouldmeetthehourlineasshownin shadow ofthehypotenuseatanygivenhourwouldfallalongline. gle withitshypotenusemakinganangle If thevertical(human)gnomonweretobereplacedwitharight-angledtrian- Intersection oftheshadowswithhourlines 1.00 mto1.90areshowninFigure14. tion 2.32mfromthecentreofsundial.Thepositionsforheights For example,forapersonofheight1.80m, GP, ofapersonstandingon thesolarclockat the tipofperson’s shadowfalls(seeFigure12). So apersonofheight1.8mwouldstandonthenorth–southlineatposi- Figure 16showstheview fromaboveofthehourline, The timeatPiazzaItalia isthereforeindicatedbythehourlineonwhich Figure 14.Relationshipbetween heightofpersonandpositionatwhichtheystand. Figure 15.Tip ofshadowhumangnomonmeetsthehourline. L with thehorizontal,then G. Thesetwolinesintersect at OP, andtheshadow, The mathematics of sundials Australian Senior Mathematics Journal 22 (1) 21 (2) is the of is the declination D is the angle hour measured - is the angle hour T is the latitude, and is the latitude, and L can be calculated according to the following can be calculated (the Sun’s ) that the shadow of a vertical the shadow of azimuth) that (the Sun’s Z can be found by solving simultaneously the equations the by solving simultaneously can be found OP: GP: Z, Figure 16. Intersection of person’s shadow with the hour line. Intersection of person’s 16. Figure . The angle Z OP. The angle and GP But tan SoEquation of shadow Substituting for cot (1) Equation of hour line From equations (1) and (2), gnomon makes with the north–south line depends on the latitude, the time depends on the north–south line makes with the gnomon and the day changes through of the Sun, which the declination of day and Angle through the . Sangwin, 2000), where formula (Budd & line, wise from the north–south for the Sun. . The coordinates of P P. The coordinates 22 Australian Senior Mathematics Journal 22 (1) Vincent 1.80 mstandingat (D tions oftheshadowtipduringdayforequinoxesandsummer Hence thecoordinatesoftipperson’s shadowwillbe: Substituting tofindthe Multiplying bothsidesbysin Figure 17.Solarishourlinesandthepositionsofshadow tip duringthedayforequinoxesand For Solaris, L = 37.8°.Figure17showsaplotofthehourlinesandposi- = 0°,23.5°and–23.5°respectively)forapersonofheight G (0, 2.32). y-coordinate oftheintersection, the summerandwintersolstices. T sin L, The mathematics of sundials Australian Senior Mathematics Journal 22 (1) 23 . : Frances Lincoln. is the at Piazza Italia. Melbourne’s time at Piazza is the solar Analemmatic Sundials: How to Build One and Why They . Oxford: Oxford University Press. Mathematics Galore Sundials: , Art, People, . Accessed August 2007 at http://plus.maths.org/issue11/features/sundials. cal orbit. a circle and the Sun is not quite at the centre of the ellipse. a circle and the Sun is not quite Work The time indicated by Solaris The time •ellipti- axis is tilted at an angle of 23.5° to the plane of the the Earth’s • than path as it revolves around the Sun is an ellipse, rather the Earth’s Solaris. Budd, C. J. & Sangwin, C. J. (2001). Lennox-Boyd, M. (2006). References Budd, C. J. & Sangwin, C. J. (2000). Corrections time to be applied to sundial to the need for correcting sundial time for Reference has been made already from the on which local time is the local longitude if that differs be adjusted slightly for two further reasons: based. Sundial time also needs to Analemmatic sundials Analemmatic which has a horizontal dial sundial is the analemmatic dial, A further type of the analemmatic sundial Like the horizontal sundial, and a vertical gnomon. by projecting the onto a is derived from an equatorial sundial projection onto the horizontal plane horizontal plane, but it is an orthogonal axis. Examples of interactive the Earth’s rather than in the direction of at the Mount Annan site of the Royal analemmatic sundials are to be found Victoria. in Torquay Botanic Gardens, Sydney and at longitude is approximately 5° west of the Australian Eastern Australian Eastern 5° west of the is approximately longitude corre- 5 degrees degrees everyEarth rotates 15 . If the (AEST) hour, is approximately time in Melbourne is, local solar 20 , that sponds to time indicated 20 minutes must be added to the so AEST, 20 minutes behind by