Weeks 1-12, 2026

The working over of Art and Mathematics in Krems, the Wachau and Borgo Sansepolcro continues at such a pace that I am unable to maintain the Reloading Humanism blog. Illustrating the progress that has been made, where a year ago I was unable to make either head nor tail of Sylvia Ronchey’s L’Enigma di Piero, I now find that I am informed enough to be able to work out what she is saying despite the fact that the book is in Italian and my Italian is rudimentary in the extreme. Happily, not only do I agree with everything that she says, but my insights support hers.

In the National Library in Vienna, I am a model for a series of publicity photos and pose in the room where I often consult manuscripts and the library’s older printed books (copyright: Österreichische Nationalbibliothk / Klaus Pichler).

Above and below I am shown looking at a facsimile of Maria of Burgundy’s illuminated prayer book.

In the music collection, we examine a hymn book used by the choir of the Imperial Court Chapel. This dates from the reign of Emperor Friederich III, who in Art and Mathematics is an important background figure as it was at his court that Peuerbach and Bessarion met, with Bessarion commissioning from the renowned Austrian astronomer, an updated commentary on Ptolemy’s Almagest.

Although Peuerbach started work immediately, he died a few months later and the task was taken on by Regiomontanus, who was Peuerbach’s star pupil. Under Bessarion’s patronage, Regiomontanus learnt Greek, translated the Almagest and then went on to write a number of works that were immensely important for the further development of astronomy and mathematics. Friedrich III also initiated the Melk Reform, a process in which the lapsed monastic ideal was revived and monasteries reasserted the importance of learning, with this prompting them to confront and get to grips with the new view of the world that science was bringing into being. It was against this background, that a theological discussion known as The Controversy of the Doctrine of Learned Ignorance unfolded, with the monastery at Melk and the Charterhouse at Aggsbach both playing important roles. Via Cusanus and his realisation that pi is a transcendental number and not a conventional irrational, the new mathematical vision of God was communicated to Piero della Francesca when he and Cusanus were both in Rome in 1458/1459. This I argue, is discernible from Piero’s Flagellation, which Ronchey argues, was commissioned by Bessarion.

Meanwhile as a virtual visitor, in the Vatican Library in Rome, among the hundreds of books that have been made available online, I find a poem written by a visitor at the Court of Urbino in which a painting by Piero della Francesca serves as the departure point for an eulogy on the Duke of Urbino, Frederigo da Montefeltro, and on the God who endowed him with a soul. This is Codex Urb. Lat. 1193, https://digivatlib.it:

A painted Image by Piero della Francesca of Burgo Sansepolcro – An Address to the God of the Same

In the poem, the portrait addresses the viewer, pointing out that even the most famed artists of Antiquity would have longed to produce an image such as Piero has produced:

By a great and awe-inspiring hand I was made:

so renowned that Zeuxis would sore feign it as his own,

in marble Euphranor count it to his oeuvre,

whilst Pyrgoteles would have made a gem or bowl.

Not the work of Praxitiles or Lysippos,

nor Polycrates or Phidias’ skilled hand,

no, it was Piero who gave me flesh, nerves and bones,

whilst you Lord, enriched me with the depths of your soul.

Speaking I strive to defend that planted within

and so as a ruler and patron, praise God’s Glory.

Although now this portrait is no longer extent, two other images of Frederigo painted by Piero della Francesca have survived, including the double portrait showing him together with his wife, Battista Sforza, which has become iconic of the Renaissance.

The work is a consolation piece commissioned so as to praise Battista following her tragic death at the age of 26. Despite Battista being married at the age of 14 to a man who was 14 older, the marriage was a happy one. At Frederigo’s court Battista studied philosophy and as Countess (for Frederigo had not yet been appointed Duke) flourished, managing her husband’s affairs when he was away on military campaigns. During their years together, the only thing that marred their happiness was the fact that Battista repeatedly boure daughters and Frederigo need a son if he was to have a direct heir. After eight daughters, the long awaited son arrived, only weakened by pregnancy, when Battista caught pneumonia some months later, she became terminally ill. Hurrying back from a campaign, Frederigo found his wife unable to speak and she died soon after. Devastated, Frederigo vowed never to marry again, gave up campaigning and as indicated by the red hat of scholarship, deoted himself to study.

Checking over the last chapter of my manuscript, I find that unplanned, but highly appropriate, the work ends with the words „out of itself“ – this being nothing other than the Reloading Humanism motto.

Yet this is not all and over the course of February, work has also proceeded apace on the missing diagrams and images so that, on the 19th March, the long awaited day arrives when I complete the last image. This is a drawing of how in around 1413, Fillipo Brunilleschi staged a demonstration that it was possible, via perspective, for artists to paint images of the world that indistinguishable from the same scene as reflected by a mirror. To this effect, standing at the entrance to the Cathedral of Santa Maria del Fiore, he painted an image of the Baptistry opposite. Then, where the central vanishing point was, he made hole. Holding a mirror out before him, in which there was likewise a hole, and with the painted image of the Baptistry facing the real Baptistry, he aligned the painting and the mirror so that through the two holes he could see the Baptistry doors. In this way what he had painted was perfectly aligned with the reality that he had depicted. As the presence and absence of the mirror made no difference to what was seen, what was painted was indistinguishable from what had been painted. To heighten the effect and to stop the weather spoiling the illusion, instead of painting the sky, Brunilleschi guilded the painting with silver, so that through the combination of painted image with a twice reflected sky, viewers were unable to tell the difference.

Week 13, 2026

In experiments conducted to ascertain whether people really do prefer the Golden Section over other proportions, although a significant amount of people do respond accordingly, there around two thirds of people who do not. Not surprisingly, this invites scepticism as to whether the ratio really does play a role in aesthetics. There is also the question of how precise is the ability of people to distinguish between two very similar proportions when for example, the rectangles that are typically used to embody the ratios are not next to each other and cannot be readily compared? Is there not a band of tolerance? Although common sense says that there must be, this is a weak weapon when it comes to countering scepticism and in any case is not scientific. Reluctant to invest time investigating statistical analysis techniques, I realise that the question is also one of information and the preservance of signals transmitted via a medium. Looking on the internet for something along these line I find an article that describes exatly what I am looking for: optimization-online.org. Thus as the week begins, I find myself incorporating the techniques into Art and Mathematics with the conjected band of tolerance underwritten by some hard-nosed maths. In the case of the rectangles used by Fechtner, where 35% of people preferred the rectangle with Golden Section proportions, 39% of people preferred either the sligtly smaller or the slightly larger rectangles that are only marginally different from the Golden Rectangle. Using information theory and statistcs, the article shows that both rectangles belong to the band of tolerance that is associated with the Golden Rectangle. In this way, preferance for the Golden Rectangle is raised from 35% to 75%.

During the week, I also finalise the „blub“ for the back cover of Art & Mathematics:

Making geometry and music central to his philosophy, Plato denied the ability of artists to depict the true nature of things. Countering this, Renaissance artists stressed the mathematical nature of perspective and the musical nature of proportion. Meanwhile anomalies in the calendar prompted astronomers to question the work of their predecessors and develop a new form of mathematics. Responding to this, Cusanus argued that God could be approached through geometry. In the Wachau this was challenged and a religious controversy ensued. Yet all the while, Piero della Francesco was using complex geometrical constructions as the basis of a new form of painting. Prompted by Cusanus, he combined all of these themes with a pressing political issue and painted The Flagellation. Using a hidden method of construction, in his Resurrection he then squared a circle. Twenty years later, in a mathematical treatise, he hinted at the geometrical key that he had used. In Krems this then provided the ground plan for a forgotten masterpiece of northern Renaissance art.

Week 14, 2026

In my reading I decide that it is time for Lesley Chamberlain’s The Philosophy Steamer, in which the story is told of how Lenin expelled Russia’s intelligensia so as to ensure that the strictly material form of communism that he envisaged would not become contaminated with rival views. This might seem a strange choice but in fact is as relevant as it is topical. For Chamberlain, the key message that Nikolai Berdyaev, repeated again and again, was that without a sense of the transcendeant, he did not see how humanity would be able to remain in touch with its greater aspirations towards spiritual freedom and moral self-determination (p. 20). A key aim of Bolshevism, Chamberlain says, was to destroy not only the Church as an automous instituition but also Christianity as a source of popular authority. Here a key objection was that religion sanctioned inwardness, with this allowing the individual freedom of thought. In the form of communisn that Lenin envisaged, a spiritual take on life was not permissible and thinking for oneself was undesirable. Accordingly, all who articulated such views were rounded up and put on two ships and deported. „Soviet totalitarism“, Chamberlain writes, meant nothing less than „denying individuals the possibility of a discrete innere life“ (p. 28). Today the trap of capitalism in it current form is that whilst freedom is promised, it is really something so abstract, that only the de-humanised entity known as homo oeconomicus can attain it and in practice, real humans (with inner lives and emotions) are left pitted against one another, to become slaves to an inhuman system of mutual and systematic exploitation. In this system, everything personal and relating to an inner realm is either negated or commercialised to become a pale and vapid echo of what it once was. Of this, the philosophy steamer described by Chamberlain and its cargo of exiled intellectuals is a premonition and a chilling metaphor for what capitalism has brought about by other means. Both capitalism and communism are ideas that, aiming at ideals, need to be filled out. The problem with both is that the take on them and the way that they have been interpretated, filled out and realised, has so far always been vehmently modernist, with not enough importance being attributed to the inner world of the self and its needs.

From the evening of Maundy Thursday until the evening mass of Easter Sunday, in Catholic countries the bells of churches go silent and only the quarter hours are marked. To call the faithful to prayer, cog rattles are used so that whilst the four days of Christ’s suffering and death are a time of limbo, there is nevertheless a sense of alarm and frenzied urgency in the air. This is the time when the whole world holds its breath to see whether the Saviour will be born again. In his De visione Dei, Cusanus advocates a withdrawal into darknes, that amid darkness a true, light might be discovered and allowed to shine and I see this as being the ultimate mystery that lies behind the Easter story of death and rebirth. Accordingly, as I do every year, I make a egg and leave it for a loved one to find on Easter Day.

Week 15, 2026

The formated manuscript of Art and Mathematics in Krems, the Wachau and Borgo Sansepolcro printed out, work begins on a final edit prior to proof-reading. Finding two passages which are too mathematical and interrupt the flow, I move them to the appendix. Elsewhere I expand a little and add an illustration.

Week 16, 2026

Walking along the banks of the Danube I find an old football and realise that, composed of 20 hexagons and 12 pentagons it is an example of an Archimedean semi-regular solid.

In geometry, where each of the five regular, so-called „Platonic Solids“, are formed from a single regular figure, the semi-regular solids are formed from two or three such figures. Discovered in Antiquity, the semi-regular solids were investigated and written about by Archimedes. Yet by the Renaissance, Archimedes‘ work had become lost and all that was known about the semi-regular solids were their basic properties and that they were thirteen in number. During the fifteenth century Piero della Francesca worked out how to construct six of the lost solids, including the truncated icosahedron.

By the middle of the sixteenth century, how the remaining semi-regular solids were to be constructed had been worked out by a number of different people, including Daniele di Barberoso who in his La Practica della Perspettiva of 1568, was able to say how all thirteen solids could be drawn using ruler and compass techniques.

The six semi-reular solids re-discovered by Piero della Francesca

During the nineteenth century, a thirteenth century prayerbook was identified as being a palimpsest, which is a  manuscript where the original text and images have been washed and scraped away so that the parchment may be re-used. Dating from the tenth century, in its original form, the book was a product of the Macedonian Renaissance and featured a number of works of Greek geometry, including Archimedes‘ lost work on the thirteen solids. In the thirteen century, the text and drawings were scraped away and the volume was turned into a prayerbook. As the scratched away letters can be recognized, during the nineteenth century, the palimpsest was identified as containing lost works of Greek geometry and in 1915, the discovery was announced in an academic journal. During a tulmultuos period that followed the First World War, the monastery where the work was found was forced to evacuate its library and the manuscript went missing. For seventy years it was lost and only re-surfaced prior to being auctioned in the 1990ies. The manuscript it transpired, had been acquired by a businessman in Paris who had hoped to sell it privately. To raise the value, fake illuminations had been painted over some of the pages. Nevetheless, despite the „enhancement“, a private sale could not be arranged and the man’s daughter had no choice but to go to an auction house and accept that the matter would go public. This resulted in the monastery to whom the manuscript had originally belonged, contesting the legality of the ownership, with the charges being however overruled. At the ensuing auction, the work was purchased by an anoymous American buyer and deposited at the Walters Museum in Baltimore so that it could be studied and conserved. Using a variety of image analysing techniques, the whole of the original text has now been recovered and Archimedes‘ work on the thirteen semi-regular solids may again once be read. See: en.wikipedia.org and archimedespalimpsest.org.

Week 17

The week begins with my starting work on a replica of a universal astrolabe designed by Georg von Peuerbach, which despite its nme, is in fact a sun dial that can be set for any desired latitude. Here the original plans of the parts are to be found on the pages of Codex Yale 24, with my sketch of the assembled whole (which dates from week 35 of 2025) being:

The first step is make the organum Ptolemei on which the scales of hours and days of the year are given.

The inner circle which encloses the organum Ptolemei is then cut out and, fitted with a sleeve from which it is supported and a moveable scale and an alidade with sighting vanes, is then theoretically, ready for use.

In practice however, the instrument requires a widget or screw system on the back so that, set to the latitude of observation, the plate may be secured to the sleeve whilst the height of the sun is being measured using the alidade. As Peuerbach gives detailed instructions on how the geometry of the instrument is to be utilised but nothing on the mechanical practicalities, the implication is that his real purpose is to show and bring to the fore, the sines and cosines that mathematically describe the position of the sun in the sky and which he can otherwise only refer to using long and cumbersome concaternations of words. The reason for this is that modern algebra was only just beginning to come into being and the use of graphs was not yet widespread.

 

graphswords.  ly interested   scale the plate to the sleeve and the scale to