Sir Lawrence Alma-Tadema’s painting *The Roses of Heliogabalus* (1888) is based on an apocryphal episode in the sybaritic life of the Roman Emperor Elagabalus (204-222 A.D.), who is said to have suffocated guests with flowers at one of his feasts. The painting is in a private collection, but I saw it for real in an Alma-Tadema exhibition at the Walker Art Gallery in Liverpool sometime during the late 1990s. I wasn’t disappointed: it was a memorable meeting with a painting I’d been interested in for years. *Roses* is impressively large and impressively skilful. Close-up, the brush-strokes are obvious, obtrusive and hard to interpret as people and objects. It isn’t till you step back, far beyond the distance at which Alma-Tadema was painting, that the almost photographic realism becomes apparent. But you get more of the many details at close range, like the Latin inscription on a bowl below and slightly to the right of that scowling water-mask. Alas, I forgot to take a note of what the inscription was, though perhaps the memory is still locked away somewhere in my subconscious.

Whatever it is, I feel sure it is significant, because *Roses* is rich with meaning. That’s a large part of why I’m interested in it. Yes, I like it a lot as art, but the women would have to be more attractive for it to be higher in the list of my favourite paintings. As it is, I think there are only four reasonably good-looking people in it: the man with the beard on the right; the flautist striding past the marble pillar on the left; the red-headed girl with a crown of white flowers; and Heliogabalus himself, crowned in roses and clutching a handful of grapes beside the overweight man who’s wearing a wreath and sardonically saluting one of the rose-pelted guests in the foreground. When I first wrote about *Roses* in a pub-zine whose name escapes me, I misidentified the overweight man as Heliogabalus himself, even though I noted that he seemed many years old than Heliogabalus, toppled as a teen tyrant, should have been. It was a bad mistake, but one that, with less knowledge and more excuse, many people must make when they look at *Roses*, because the overweight man and his sardonic salute are a natural focus for the eye. Once your eye has settled on and noted him, you naturally follow the direction of his gaze down to the man in the foreground, who’s gazing right back.

And by following that gaze, you’ve performed a little ratio-ritual, just as Alma-Tadema intended you to do. Yes, *Roses* is full of meaning and much of that meaning is mathematical. I think the angle of the gaze is one of many references in *Roses* to the golden ratio, or φ (phi), a number that is supposed to have special aesthetic importance and has certainly been used by many artists and musicians to guide their work. A rectangle with sides in the proportions 8:13, for example, approximates the golden ratio pretty closely, but φ itself is impossible to represent physically, because it’s an irrational number with infinitely many decimal digits, like π or √2, the square root of two. π represents the ratio of a circle’s circumference to its diameter and √2 the ratio of a square’s diagonal to its side, but no earthly circle and no earthly square can ever capture these numbers with infinite precision. Similarly, no earthly rectangle can capture φ, but the rectangle of *Roses* is a good attempt, because it measures 52″ x 84 ^{1}/_{8}". That extra eighth of an inch was my first clue to the painting’s mathematical meaningfulness. And sure enough, 52/84·125 = 416/673 = 0·61812…, which is a good approximation to φ’s never-ending 0·6180339887498948482045868343656…*

That deliberate choice of dimensions for the canvas led me to look for more instances of φ in the painting, though one of the most important and obvious might be called a meta-presence. *The Roses of Heliogabalus* is dated 1888, or 1666 years after the death of Heliogabalus in 222 AD. A radius at 222º divides a circle in the golden ratio, because 222/360 = 0·616… It’s very hard to believe Alma-Tadema didn’t intend this reference and I also think there’s something significant in 1888 itself, which equals 2 x 2 x 2 x 2 x 2 x 59 = 2^{5} x 59. Recall that 416 is the expanded short side of *Roses.* This equals 2^{5} x 13, while 673, the expanded long side, is the first prime number after 666. As one of the most technically skilled painters who ever lived, Alma-Tadema was certainly an exceptional implicit mathematician. But he clearly had explicit mathematical knowledge too and this painting is a phi-pie cooked by a master matho-chef. In short, when *Roses* is read, *Roses* turns out to be golden.

*φ is more usually represented as 1·6180339887498948482045868343656…, but it has the pecularity that ^{1}/_{φ} = φ-1, so the decimal digits don’t change and 0·6180339887498948482045868343656… is also legitimate.

**Appendix I**

I’ve looked at more of Alma-Tadema’s paintings to see if their dimensions approximate φ, √2, √3 or π, or their reciprocals. These were the results (ε = error, i.e. the difference between the constant and the ratio of the dimensions).

*The Roman Wine Tasters* (1861), 50" x 69 ^{2}/_{3}": 150/209 = 0·717… ≈ ^{1}/_{√2} (ε=0·02)

*A Roman Scribe* (1865), 21 ^{1}/_{2}" x 15 ^{1}/_{2}": 43/31 = 1·387… ≈ √2 (ε=0·027)

*A Picture Gallery* (1866), 16 ^{1}/_{8}" x 23": 129/184 = 0·701… ≈ ^{1}/_{√2} (ε=0·012)

*A Roman Dance* (1866), 16 ^{1}/_{8}" x 22 ^{1}/_{8}": 43/59 = 0·728… ≈ ^{1}/_{√2} (ε=0·042)

*In the Peristyle* (1866), 23" x 16": 23/16 = 1·437… ≈ √2 (ε=0·023)

*Proclaiming Emperor Claudius* (1867), 18 ^{1}/_{2}" x 26 ^{1}/_{3}": 111/158 = 0·702… ≈ ^{1}/_{√2} (ε=0·009)

*Phidias and the Frieze of the Parthenon Athens* (1868), 29 ^{2}/_{3}" x 42 ^{1}/_{3}": 89/127 = 0·7… ≈ ^{1}/_{√2} (ε=0·012)

*The Education of Children of Clovis* (1868), 50" x 69 ^{2}/_{3}": 150/209 = 0·717… ≈ ^{1}/_{√2} (ε=0·02)

*An Egyptian Juggler* (1870), 31" x 19 ^{1}/_{4}": 124/77 = 1·61… ≈ φ (ε=0·007)

*A Roman Art Lover* (1870), 29" x 40": 29/40 = 0·725… ≈ ^{1}/_{√2} (ε=0·034)

*Good Friends* (1873), 4 ^{1}/_{2}" x 7 ^{1}/_{4}": 18/29 = 0·62… ≈ φ (ε=0·006)

*Pleading* (1876), 8 ^{1}/_{2}" x 12 ^{3}/_{8}": 68/99 = 0·686… ≈ ^{1}/_{√2} (ε=0·041)

*An Oleander* (1882), 36 ^{1}/_{2}" x 25 ^{1}/_{2}": 73/51 = 1·431… ≈ √2 (ε=0·017)

*Dolce Far Niente* (1882), 9 ^{1}/_{4}" x 6 ^{1}/_{2}": 37/26 = 1·423… ≈ √2 (ε=0·008)

*Anthony and Cleopatra* (1884), 25 ^{3}/_{4}" x 36 ^{1}/_{3}": 309/436 = 0·708… ≈ ^{1}/_{√2} (ε=0·003)

*Rose of All Roses* (1885), 15 ^{1}/_{4}" x 9 ^{1}/_{4}": 61/37 = 1·648… ≈ φ (ε=0·03)

*The Roses of Heliogabalus* (1888), 52" x 84 ^{1}/_{8}": 416/673 = 0·618… ≈ φ (ε<0.001)

*The Kiss* (1891), 18" x 24 ^{3}/_{4}": 8/11 = 0·727… ≈ ^{1}/_{√2} (ε=0·039)

*Unconscious Rivals* (1893), 17 ^{3}/_{4}" x 24 ^{3}/_{4}": 71/99 = 0·717… ≈ ^{1}/_{√2} (ε=0·019)

*A Coign of Vantage* (1895), 25 ^{1}/_{4}" x 17 ^{1}/_{2}": 101/70 = 1·442… ≈ √2 (ε=0·028)

*A Difference of Opinion* (1896), 15" x 9": 5/3 = 1·666… ≈ φ (ε=0·048)

*Whispering Noon* (1896), 22" x 15 ^{1}/_{2}": 44/31 = 1·419… ≈ √2 (ε=0·005)

*Her Eyes Are With Her Thoughts And Her Thoughts Are Far Away* (1897), 9" x 15": 3/5 = 0·6… ≈ φ (ε=0·048)

*The Baths of Caracalla* (1899), 60" x 37 ^{1}/_{2}": 8/5 = 1·6… ≈ φ (ε=0·018)

*The Year’s at the Spring, All’s Right with the World* (1902), 13 ^{1}/_{2}" x 9 ^{1}/_{2}": 27/19 = 1·421… ≈ √2 (ε=0·006)

*Ask Me No More* (1906), 31 ^{1}/_{2}" x 45 ^{1}/_{2}": 9/13 = 0·692… ≈ ^{1}/_{√2} (ε=0·03)

**Appendix II**

*The Roses of Heliogabalus* is based on this section from Aelius Lampridius’ pseudonymous and largely apocryphal *Vita Heliogabali*, or *Life of Heliogabalus*, in the *Historia Augusta* (late fourth century):

XXI. 1 Canes iecineribus anserum pavit. Habuit leones et leopardos exarmatos in deliciis, quos edoctos per mansuetarios subito ad secundam et tertiam mensam iubebat accumbere ignorantibus cunctis, quod exarmati essent, ad pavorem ridiculum excitandum. 2 Misit et uvas Apamenas in praesepia equis suis et psittacis atque fasianis leones pavit et alia animalia. 3 Exhibuit et sumina apruna per dies decem tricena cottidie cum suis vulvis, pisum cum aureis, lentem cum cerauniis, fabam cum electris, orizam cum albis exhibens. 4 Albas praeterea in vicem piperis piscibus et tuberibus conspersit. 5 Oppressit in tricliniis versatilibus parasitos suos violis et floribus, sic ut animam aliqui efflaverint, cum erepere ad summum non possent. 6 Condito piscinas et solia temperavit et rosato atque absentato…

XXI. 1 He fed his dogs on goose-livers. He had pet lions and leopards, which had been rendered harmless and trained by tamers, and these he would suddenly order during the dessert and the after-dessert to get on the couches, thereby causing laughter and panic, for none knew that they were harmless. 2 He sent grapes from Apamea to his stables for the horses, and he fed parrots and pheasants to his lions and other beasts. 3 For ten days in a row, moreover, he served wild sows’ udders with the matrices, at a rate of thirty a day, serving, besides, peas with gold-pieces, lentils with onyx, beans with amber, and rice with pearls; 4 and he also sprinkled pearls on fish and used truffles instead of pepper. 5 In a banqueting-room with a reversible ceiling he once buried his parasites in violets and other flowers, so that some were actually smothered to death, being unable to crawl out to the top. 6 He flavoured his swimming-pools and bath-tubs with essence of spices or of roses or wormwood…