"… I seem […] only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me". – Isaac Newton.
AGGIORNAMENTO: Nel maggio 2025, il progetto ha ottenuto riconoscimento vincendo il 1° premio nel concorso Instructables All Things Pi. Un grande grazie al team di Instructables!
Qualche giorno fa ho pubblicato un nuovo progetto educativo sul mio sito Instructables. Il dispositivo, che ho battezzato RaPenduLa (dalle iniziali in inglese di RaspPi Pendulum Laboratory), è stato ribattezzato in italiano CAMPO (Computer Analisi Moto Pendolare Oscillante) grazie a un suggerimento di ChatGPT. Ma, come direbbe Shakespeare, ‘What’s in a name? That which we call a rose by any other name would smell as sweet’: il cuore del progetto è infatti una piattaforma video per lo studio delle oscillazioni meccaniche. Utilizzando un Raspberry Pi Zero W2 dotato di modulo fotocamera, il sistema registra ad alta velocità il movimento dei pendoli. Poi, con un’analisi video basata su Python e OpenCV, RaPenduLa è in grado di tracciare il percorso preciso della punta del pendolo, visualizzandone il comportamento oscillatorio in 2D.
UPDATE: In May 2025, the project achieved recognition by winning 1st prize in the Instructables contest All Things Pi. A big thank you to the Instructables teams!
I have recently published another educational project on my Instructables website. I called the device RaPenduLa for the RaspPi Pendulum Laboratory, and it is a video platform for studying mechanical oscillations. It uses a Raspberry Pi Zero W2 equipped with a camera module to record the motion of pendulums in high speed. The interesting part happens through video analysis: using Python and the fantastic OpenCV library, RaPenduLa can track the precise path of a pendulum’s tip and help visualize its oscillatory behavior in two dimensions.
It has become a recurrent habit for me to write a blog on the shape of eggs to wish you a Happy Easter. Not repeating oneself and finding a new interesting topic is a brainstorming exercise of lateral thinking and a systematic search in literature to find an interesting connection. This year, I wanted to explore an idea that has been lurching in my mind for some time for other reasons: billiards.
I used to play snooker from time to time with some old friends. I am a far cry from being even an amateur in the billiard games, but I had a lot of fun verifying the laws of mechanics on a green table. I soon discovered that studying the dynamics of bouncing collision of an ideal cue ball in billiards of different shapes keeps brilliant mathematicians and physicists engaged in recreational academic studies and important theoretical implications.
My heart leaps up when I behold A rainbow in the sky: So was it when my life began; So is it now I am a man; So be it when I shall grow old, Or let me die! The Child is father of the Man; And I could wish my days to be Bound each to each by natural piety.
William Wordsworth, March 26, 1802
I couldn’t resist citing the beautiful poetry by Wordsworth about the rainbow to introduce my new Instructable, ‘Explore the Physics of Soap Films with the SoapFilmScope.’ I got the idea for this project by reading an article by Gaulon et al. [1]. The authors describe in detail the use of soap film as an educational aid to explore interesting effects in the fluid dynamics of this system. In particular, they examine the impact of acoustic waves on the unique optical properties of the film. In this Instructable, we have designed a device called the SoapFilmScope to perform these experiments. This tutorial will guide you through the process of creating this device, showcasing the mesmerizing interaction between sound waves and liquid membranes. The SoapFilmScope offers an engaging way to explore the physics of acoustics, light interference, and fluid dynamics.
When a sound wave travels through the tube and vibrates the soap film, it creates dynamic patterns through several fascinating mechanisms:
The device consists of a vertical soap film delicately suspended at the end of a tube obtained from a PVC T-shaped fitting that you can get from any DIY store. By attaching a small inexpensive speaker to it, you can let the film dance to the rhythm of the music.
Sometime ago, I have written about the Lissajous-Bowditch figures. In the same article, it is described how to build a simple device called a kaleidophone to generate Lissajous patterns. Using a small mirror fixed securely to the end of a bent wire on a stable platform and a laser beam from a laser pointer reflects off it, mesmerizing, intertwined spirals of light. The laser beam will appear dancing on the wall of your room. This enchanting display results from two mutually perpendicular harmonic oscillations generated by the vibrations of the elastic wire. These captivating patterns are known as Lissajous-Bowditch figures and are named after the French physicist Jules Antoine Lissajous, who did a detailed study of them (published in his Mémoire sur l’étude optique des mouvements vibratoires, 1857). The American mathematician Nathaniel Bowditch (1773 – 1838) conducted earlier and independent studies on the same curves, and for this reason, the figures are also called Lissajous-Bowditch curves [2].
LB curves result from the combination of two harmonic motions, and therefore, they can be mathematically generated through a parametric representation involving two sinusoidal functions (see Figure and also here). Lissajous invented different mechanical devices reproducing these periodic oscillations consisting of two mirrors attached to two oriented diapasons (or other oscillators) by double reflecting a collimated ray of light on a screen, producing these figures upon oscillations of the diapasons. The diapason can be substituted with elastic wires, speakers, pendulum, or electronic circuits. In the last case, the light is the electron beam of a cathodic tube (or its digital equivalent) of an oscilloscope [3].
The simplest of these devices is the KaleidoPhone, invented (and named) by the British physicist Charles Wheatstone at the beginning of the 19th century [3,4]. The Kaleidophone creates stunning Lissajous patterns and is an excellent example of how science can also be an art form. You can bring the mesmerizing dance of light to life with just a few simple materials and creativity.
In a new Instructable project, I have presented a modern compact version of the kalidophone device fabricated with the help of 3D printing technology and enhanced with a digital camera.
For this last bit of modern technology, the new device is called KaleidoPhoneScope. What makes this little device is the facility to adapt it to record another form of vibrations by adding a speaker and another mirror free to vibrate on its bizarre pattern, recalling SciFi movies promp appear on a free wall (or door) of your studio.
As Christmas approaches, what is the best time to try this device with a traditional song? Here is the result. Activate the captions to see the corresponding frequencies of the tones.
I wish you all to spend a Merry Christmas with your dearest, and I hope to see a peace and
REFERENCES
T. B. Greenslade Jr., “All about Lissajous figures,” The Physics Teacher, 31, 364 (1993).
T. B. Greenslade Jr., “Devices to Illustrate Lissajous Figures,” *The Physics Teacher, 41, 351 (2003).
C. Wheatstone, Description of the kaleidophone, or phonic kaleidoscope: A new philosophical toy, for the illustration of several interesting and amusing acoustical and optical phenomena, Quarterly Journal of Science, Literature and Art 23, 344 (1827).
R. J. Whitaker, “The Wheatstone kaleidophone,” American Journal of Physics, 61, 722 (1993).
Alatri is a picturesque town in the province of Frosinone, 80 km southeast of Rome. Located in the heart of Ciociaria, it overlooks the Sacco Valley from the highest point on top of a hill. Among other important attractions, the historical center of the city treasures one of the best-preserved megalithic constructions in Ciociaria. Megalithic architecture is characterized by imposing stonework (also called megalithic or Cyclopean ruins). The one in Alatri is an impressive perimeter wall. The wall, known as the polygonal walls, is built around the highest point of the town, the acropolis. It was constructed for fortification and ritual worship purposes. The megalithic civilization reached the highest level of stone masonry during the Neolithic period. Major centers existed in different Italian regions and throughout Europe from Greece to England. Alatri is just one of the megalithic towns of Ciociaria. Other nearby impressive remains can be found in Ferentino and Veroli.
I will write more about megalithic architecture in another article. In this context, I will describe a more recent but still beautiful architectural embellishment. This embellishment has a practical function. It is prominently visible in the Piazza Santa Maria Maggiore, the town’s central square. I am referring to the beautiful sundial (OROLOGIO SOLARE in italiano, see photo below). It was built in 1867 on the facade of the Palazzo Conti Gentili. The architect Giuseppe Olivieri constructed it based on accurate calculations by Padre Angelo Secchi. He was a renowned Jesuit and astronomer. A photo of the sundial is reported below. In Italian, the sundial is translated as orologio solare, as written in the image.
Figure 1: Photo of the Secchi’s sundial located in Piazza Santa Maria Maggiore of Alatri.
The analysis of this sundial gave me the opportunity to introduce the principles used to build it. I also learned a bit more about astronomical calculations. Therefore, I want to share with the reader my findings.
What is a sundial?
The sundial (also called meridian) is a time-measuring device based on the regular rotation of the Earth. The Sun’s apparent position in the sky changes the shadow’s projection cast by the dial. This shadow falls on a surface that has been time marked. As a result, the surface can have different orientations and shapes. The Secchi’s sundial is a vertical type with orientation North-South. The title on the top states this: The Secchi’s sundial shows the real time. It also shows the average time (OROLOGIO SOLARE A TEMPO VERO E MEDIO). The calligraphic text on the bottom indicates the geographic coordinates of the sundial.
The latitude and longitude indicate the location of the bell tower. It is part of the cathedral of Alatri (duomo di Alatri o Basilica di San Paolo). As reference meridian (the prime meridian) was consider the one passing for the city of Rome. In particular, it is the meridian that passes through the Collegio Romano observatory. Secchi was the director there at the time of the sundial’s construction. The international adoption of the prime meridian passing through London was agreed upon during an international geographic conference. This conference was organized in the same city in October 1884. Before this date, country were used to adopt their own prime meridian, usually passing for the capital. So it is not surprising that Sacchi used as reference meridian the one passing for Rome. The Colleggio Romano was a school established by founder of the Society of Jesus St. Ignatius of Loyola in 1551. It is located in the Piazza del Collegio Romano in the Pigna District. P.A. Secchi was the director of the astronomic observatory of the school. The Monte Mario Observatory was constructed in 1934, at Villa Mellini. This moved the prime meridian for Rome there. It was used as the reference meridian for Italy’s geographic maps until 1960.
The geographic coordinates of the cathedral of Alatri given by Google Maps are 41.7248° N, 13.3443° E. Therefore, Sacchi approximated the longitude to the one of the Collegio Romano (41.8988° N, 12.4807° E) that he could accurately calculate. According to Google Earth, the sundial’s position is 41°43′ 35.86″ N, 13° 20′ 33.86″ E. Therefore, the prime meridian is used to calculate the real-time of the sundial.
Figure 2: Position of the Piazza Santa Maria Maggiore and of the cathedral of Alatri (bottom complex). Source Google Earth.
The length of the shadow cast by the sundial varies with the Sun’s altitude, and it also changes during the year as the Earth moves along an orbit that is inclined by ~23.4° concerning the ecliptic plane (the position of the Sun’s equator). The length defines a particular position for the Earth in its orbit, as the solstices and equinoxes are the dates in between. The length of the shadow is marked on the solar clock with seven declination arcs. The latter ones go from left to right, delimited by the Zodiac signs and solstices, and equinoctial dates. Using the Zodiac sign is a convenient way to divide into 12 sectors of 30° the ecliptic longitude along the Earth’s orbit. This leads to 7 arcs, five crossed twice by the Sun (when its declination is increasing and decreasing), plus two for solstices (extreme declinations). As Sun’s altitude varies between +/- 23° 26′, it is also possible to draw arcs every 5° of declination, with the equinoctial line (March 21st and September 22nd) in the middle which corresponds to 0° of declination (Sun on the equator).
Description of the Components of the Sundial
The Secchi’s sundial in Alatri consists of several key components that contribute to its functionality and accuracy in measuring time. These components include the gnomon, the dial plate, the hour lines, and the declination arcs. The dial plate serves as the surface upon which the shadow of the gnomon falls. It is typically a flat, horizontal surface with markings or engravings that denote the hours of the day. In the case of this sundial, the dial plate features hour lines and declination arcs that aid in reading the time and understanding the position of the Sun.
Gnomon: The gnomon is a crucial element of the sundial, responsible for casting the shadow that indicates the time. In the case of the Secchi’s sundial, the gnomon is a vertical structure aligned in a North-South direction. It is designed to be perpendicular to the dial plate and is carefully positioned to ensure the accuracy of the shadow projection.
Hour Lines: The hour lines on a vertical sundial represent the hours of the day and are usually spaced evenly on the dial plate. To determine the position of the hour lines, we consider the angle between the gnomon and the noon line (the line pointing directly towards the Sun at solar noon). Let’s denote this angle as θ. Assuming that the gnomon is aligned perfectly North-South, the angle θ can be calculated using the latitude of the sundial’s location (represented by φ). The equation is as follows: Once we have the angle , we can divide it by (since there are 15 degrees of longitude per hour) to determine the angular distance between each hour line.This angular distance can then be translated into linear distances on the dial plate based on the specific design of the sundial. In Secchi’s sundial, the lines are indicated with Roman numerals from left to right as X, XI, XII, I, II, III, IV, respectively.
Declination Arcs: The sundial incorporates declination arcs, which are curved lines that represent the lengths of the shadow at specific times of the year. In the case of the Secchi’s sundial, there are seven declination arcs. These arcs are marked by the signs of the Zodiac and the solstices and equinoxes. The declination arcs provide additional reference points for understanding the position of the Sun and the corresponding time. The declination arcs on the sundial represent the lengths of the shadow cast by the gnomon at specific times of the year. The position of these arcs can be determined using the latitude of the sundial’s location and the declination of the Sun at various points throughout the year. The declination (represented by δ) is the angular distance between the Sun and the celestial equator. It varies throughout the year due to the tilt of the Earth’s axis. The formula for calculating the declination at a given day of the year is complex and involves astronomical calculations, but it can be approximated using simpler methods. Using the simplified approximation, we can calculate the declination (δ) based on the day number (represented by n, with n=1 being January 1st). The equation is as follows: With the declination (\delta) determined, we can mark the corresponding declination arc on the sundial. The length of the arc will depend on the specific design and scale of the sundial.
Analemmas: The analemmas on a sundial are curves that represent the changing position of the Sun in the sky throughout the year. The shape of the analemma is influenced by the combined effect of the Earth’s elliptical orbit and axial tilt. The equation of time is directly related to the shape and position of the analemmas. The equation of time is a mathematical expression that represents the difference between solar time (based on the Sun’s actual position in the sky) and mean solar time (based on a uniform 24-hour day). This difference arises from two main factors: the Earth’s elliptical orbit and its axial tilt. The equation of time (EoT) is given as a function of the day of the year (n) and is typically measured in minutes. It can be positive or negative, indicating whether solar time is ahead or behind mean solar time, respectively. The equation of time affects the position of the Sun in the sky and, consequently, the position of the hour lines and analemmas on a sundial. The analemmas help to correct the discrepancy between solar time and mean solar time by providing reference points on the sundial.
By combining the gnomon, the dial plate with hour lines, and the declination arcs, the Secchi’s sundial allows for the accurate measurement of time-based on the position and length of the shadow cast by the gnomon. Observers can align the shadow with the hour lines to determine the time of day. At the same time, the declination arcs provide insights into the Sun’s position along the ecliptic throughout the year.
It is worth noting that the accuracy of the sundial’s measurements can be influenced by factors such as the precise alignment of the gnomon, the dial plate’s orientation, and the location’s latitude. However, the design of the Secchi’s sundial, with its North-South alignment and inclusion of declination arcs, enhances its accuracy and usefulness as a time-measuring device in Alatri.
Once we have the angle 
, we can divide it by 
(since there are 15 degrees of longitude per hour) to determine the angular distance between each hour line.This angular distance can then be translated into linear distances on the dial plate based on the specific design of the sundial. In Secchi’s sundial, the lines are indicated with Roman numerals from left to right as X, XI, XII, I, II, III, IV, respectively.
With the declination (\delta) determined, we can mark the corresponding declination arc on the sundial. The length of the arc will depend on the specific design and scale of the sundial.
daniloroccatano.blog 