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Oxford, 2006

In 2006, I had the privilege of attending two conferences in England (the 11^{th} International Command & Control Research & Technology Symposium in Cambridge and the Complex Adaptive Systems and Interacting Agents Workshop in Oxford).

This was the year that NASA launched the *New Horizons* spaceprobe towards Pluto (it was to arrive in 2015). Ironically, later in 2006, the International Astronomical Union somewhat controversially downgraded the status of Pluto to that of a “dwarf planet.”

Grigori Perelman’s proof of the Poincaré conjecture was declared the “Breakthrough of the Year” by the journal *Science*. A variety of books, such as this one, have tried to explain what the conjecture (now theorem) is about. So far, this is the only one of the seven Millennium Prize Problems to be solved.

Perelman was offered, but refused, the prestigious Fields Medal (in interviews, he raised some ethical concerns regarding the mathematical community).

Books of 2006 included the intriguing *World War Z* (later made into a mediocre film). Movies included *Pan’s Labyrinth*, *Children of Men*, *Apocalypto*, *Black Book*, *Pirates of the Caribbean II*, *Cars*, and *The Nativity Story*.

And in music, Carrie Underwood took the world by storm, singing about Jesus and about smashing up motor vehicles with baseball bats.

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From around 2000 BC, the Babylonians used a positional number system with base 60. Initially a space was used to represent zero. Vertical wedges mean 1, and chevrons mean 10:

This number (which we can write as 2 ; 0 ; 13) means 2 × 3600 + 0 × 60 + 13 = 7213. Four thousand years later, we still use the same system when dealing with angles or with time: 2 hours, no minutes, and 13 seconds is 7213 seconds.

Later, the Babylonians introduced a variety of explicit symbols for zero. By 400 BC, a pair of angled wedges was used:

The Babylonian zero was never used at the **end** of a number. The Babylonians were happy to move the decimal point (actually, “sexagesimal point”) forwards and backwards to facilitate calculation. The number ½, for example, was treated the same as 30 (which is half of 60). In much the same way, 20^{th} century users of the slide rule treated 50, 5, and 0.5 as the same number. What is 0.5 ÷ 20? The calculation is done as 5 ÷ 2 = 2.5. Only at the end do you think about where the decimal point should go (0.025).

Kaplan says about zero that “the Greeks had no word for it.” Is that true?

Much of Greek mathematics was done in words. For example, the famous Proposition 3 in the *Measurement of a Circle* (Κύκλου μέτρησις) by Archimedes reads:

Παντὸς κύκλου ἡ περίμετρος τῆς διαμέτρου τριπλασίων ἐστί, καὶ ἔτι ὑπερέχει ἐλάσσονι μὲν ἤ ἑβδόμῳ μέρει τῆς διαμέτρου, μείζονι δὲ ἢ δέκα ἑβδομηκοστομόνοις.

Phonetically, that is:

*Pantos kuklou hē perimetros tēs diametrou triplasiōn esti, kai eti huperechei elassoni men ē hebdomō merei tēs diametrou, meizoni de ē deka hebdomēkostomonois.*

Or, in English:

The perimeter of every circle is triple the diameter plus an amount less than one seventh of the diameter and greater than ten seventy-firsts.

In modern notation, we would express that far more briefly as ^{10}/_{71} < π − 3 < ^{1}/_{7} or 3.141 < π < 3.143.

The Greek words for zero were the two words for “nothing” – μηδέν (*mēden*) and οὐδέν (*ouden*). Around 100 AD, Nicomachus of Gerasa (Gerasa is now the city of Jerash, Jordan), wrote in his *Introduction to Arithmetic* (Book 2, VI, 3) that:

οὐδέν οὐδενί συντεθὲν … οὐδέν ποιεῖ (*ouden oudeni suntethen … ouden poiei*)

That is, zero (nothing) can be added:

nothing and nothing, added together, … make nothing

However, we cannot **divide** by zero. Aristotle, in Book 4, Lectio 12 of his *Physics* tells us that:

οὐδὲ τὸ μηδὲν πρὸς ἀριθμόν (*oude to mēden pros arithmon*)

That is, ^{1}/_{0}, ^{2}/_{0}, and so forth make no sense:

there is no ratio of zero (nothing) to a number

If we view arithmetic primarily as a game of multiplying, dividing, taking ratios, and finding prime factors, then poor old zero really does have to sit on the sidelines (in modern terms, zero is not part of a multiplicative group).

For business calculations, surveying, numerical tables, and most other mathematical calculations (e.g. the proof of Archimedes’ Proposition 3), the Greeks used a non-positional decimal system, based on 24 letters and 3 obsolete letters. In its later form, this was as follows:

Units | Tens | Hundreds |
---|---|---|

α = 1 | ι = 10 | ρ = 100 |

β = 2 | κ = 20 | σ = 200 |

γ = 3 | λ = 30 | τ = 300 |

δ = 4 | μ = 40 | υ = 400 |

ε = 5 | ν = 50 | φ = 500 |

ϛ (stigma) = 6 | ξ = 60 | χ = 600 |

ζ = 7 | ο = 70 | ψ = 700 |

η = 8 | π = 80 | ω = 800 |

θ = 9 | ϙ (koppa) = 90 | ϡ (sampi) = 900 |

For users of R:

```
to.greek.digits <- function (v) { # v is a vector of numbers
if (any(v < 1 | v > 999)) stop("Can only do Greek digits for 1..999")
else {
s <- intToUtf8(c(0x3b1:0x3b5,0x3db,0x3b6:0x3c0,0x3d9,0x3c1,0x3c3:0x3c9,0x3e1))
greek <- strsplit(s, "", fixed=TRUE)[[1]]
d <- function(i, power=1) { if (i == 0) "" else greek[i + (power - 1) * 9] }
f <- function(x) { paste0(d(x %/% 100, 3), d((x %/% 10) %% 10, 2), d(x %% 10)) }
sapply(v, f)
}
}
```

For example, the “number of the beast” (666) as written in Byzantine manuscripts of the Bible is χξϛ (older manuscripts spell the number out in words: ἑξακόσιοι ἑξήκοντα ἕξ = *hexakosioi hexēkonta hex*).

This Greek system of numerals did not include zero – but then again, it was used in situations where zero was not needed.

Most of Greek mathematics was geometric in nature, rather than based on calculation. For example, the famous Pythagorean Theorem tells us that the areas of two squares add up to give the area of a third.

In geometry, zero was represented as a line of zero length (i.e. a point) or as a rectangle of zero area (i.e. a line). This is implicit in Euclid’s first two definitions (σημεῖόν ἐστιν, οὗ μέρος οὐθέν = a point is that which has no part; γραμμὴ δὲ μῆκος ἀπλατές = a line is breadthless length).

In the Pythagorean Theorem, lines are multiplied by themselves to give areas, and the sum of the two smaller areas gives the third (image: Ntozis)

In astronomy, the Greeks continued to use the Babylonian sexagesimal system (much as we do today, with our “degrees, minutes, and seconds”). Numbers were written using the alphabetic system described above, and at the time of Ptolemy, zero was written like this (appearing in numerous papyri from 100 AD onwards, with occasional variations):

For example, 7213 seconds would be β ō ιγ = 2 0 13 (for another example, see the image below). The circle here may be an abbreviation for οὐδέν = nothing (just as early Christian Easter calculations used N for Nulla to mean zero). The overbar is necessary to distinguish ō from ο = 70 (it also resembles the overbars used in sacred abbreviations).

This use of a circle to mean zero was passed on to the Arabs and to India, which means that our modern symbol 0 is, in fact, Graeco-Babylonian in origin (the contribution of Indian mathematicians such as Brahmagupta was not the introduction of zero, but the theory of negative numbers). I had not realised this before; from now on I will say *ouden* every time I read “zero.”

Part of a table from a French edition of Ptolemy’s

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Four representative solar car team activities in the lead-up to the World Solar Challenge in October –

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We have had a few new solar car reveals recently (see above – click to zoom). The pie chart below shows current statistics (excluding #67 Golden State and #86 Dyuti, which do not seem to be active teams). Among the Challengers, the designs for #4 Antakari, #10 Tokai, and #18 EcoPhoton are still unknown.

Monohulls remain a minority among the Challengers (though a minority that has doubled in size since 2017). I am using the term “outrigger” for cars with monohull bodies but wheels sticking well out to the sides (the two new Swedish teams, #23 HUST and #51 Chalmers). There are also two quite different wide symmetric cars (#22 MDH and #63 Alfaisal). Among the Cruisers, 4-seaters remain a minority, in spite of the substantial points benefit for carrying multiple passengers. As always, see my regularly updated illustrated teams list for details.

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Our first inscription was inscribed at the site of the Battle of Thermopylae (480 BC), where Leonidas and his 300 Spartans (plus several thousand allies) died trying to hold off a vastly superior Persian army. The inscription no longer exists (though there is a modern copy at the site), but the wording has been preserved by Herodotus (*Histories* 7.228.2):

Ω ΞΕΙΝ ΑΓΓΕΛΛΕΙΝ

ΛΑΚΕΔΑΙΜΟΝΙΟΙΣ ΟΤΙ ΤΗΔΕ

ΚΕΙΜΕΘΑ ΤΟΙΣ ΚΕΙΝΩΝ

ΡΗΜΑΣΙ ΠΕΙΘΟΜΕΝΟΙ.

Phonetically, that reads:

Ō ksein’, angellein

Lakedaimoniois hoti tēide

keimetha, tois keinōn

rhēmasi peithomenoi.

I’ve always thought that there was a degree of sarcasm in this laconic epigram – after all, the Spartans had declared war on the Persians (rather informally, by throwing the Persian ambassadors down a well), but then stayed home, leaving Leonidas and his personal honour guard (plus the allies) to do the actual fighting. My (rather free) personal translation would therefore be:

Go tell the Spartans,

Stranger passing by,

We listened to their words,

And here we lie.

The rich history of the Rosetta Stone has always fascinated me (and I made a point of seeing the Stone when I visited the British Museum). The Stone records a decree of 196 BC from Ptolemy V, inscribed using three forms of writing – Egyptian hieroglyphs, Egyptian demotic script, and a Greek translation. The Stone was therefore a valuable input to the eventual decoding of Egyptian hieroglyphs. Romance practically drips off the Stone.

The Theodotus inscription in Jerusalem was located in a 1^{st} century synagogue near the Temple (this dating is generally accepted). It reads as follows (with [square brackets] denoting missing letters):

ΘΕΟΔΟΤΟΣ ΟΥΕΤΤΕΝΟΥ ΙΕΡΕΥΣ ΚΑΙ

ΑΡΧΙΣΥΝΑΓΩΓΟΣ ΥΙΟΣ ΑΡΧΙΣΥΝ[ΑΓΩ]–

Γ[Ο]Υ ΥΙΟΝΟΣ ΑΡΧΙΣΥΝ[Α]ΓΩΓΟΥ ΩΚΟ-

ΔΟΜΗΣΕ ΤΗΝ ΣΥΝΑΓΩΓ[Η]Ν ΕΙΣ ΑΝ[ΑΓ]ΝΩ-

Σ[Ι]Ν ΝΟΜΟΥ ΚΑΙ ΕΙΣ [Δ]ΙΔΑΧΗΝ ΕΝΤΟΛΩΝ ΚΑΙ

ΤΟΝ ΞΕΝΩΝΑ ΚΑ[Ι ΤΑ] ΔΩΜΑΤΑ ΚΑΙ ΤΑ ΧΡΗ-

Σ[Τ]ΗΡΙΑ ΤΩΝ ΥΔΑΤΩΝ ΕΙΣ ΚΑΤΑΛΥΜΑ ΤΟΙ-

Σ [Χ]ΡΗZΟΥΣΙΝ ΑΠΟ ΤΗΣ ΞΕ[Ν]ΗΣ ΗΝ ΕΘΕΜΕ-

Λ[ΙΩ]ΣΑΝ ΟΙ ΠΑΤΕΡΕΣ [Α]ΥΤΟΥ ΚΑΙ ΟΙ ΠΡΕ-

Σ[Β]ΥΤΕΡΟΙ ΚΑΙ ΣΙΜΩΝ[Ι]ΔΗΣ.

In translation:

Theodotus, son of Vettenus [or, of the gens Vettia], priest and

archisynagogue [leader of the synagogue], son of an archisynagogue,

grandson of an archisynagogue, built

the synagogue for the reading of

the Law and for teaching the commandments;

also the hostel, and the rooms, and the water

fittings, for lodging

needy strangers. Its foundation was laid

by his fathers, and the

elders, and Simonides.

The inscription is interesting in a number of ways. Along with other similar inscriptions, it demonstrates the existence of Greek-language synagogues in 1^{st} Palestine. The title ἀρχισυνάγωγος (*archisynagōgos*) also occurs in the New Testament (nine times, starting at Mark 5:22), so is clearly a title of the time-period. Some scholars have suggested that Theodotos was a freed slave, who had made his fortune and returned from Italy to the land of his fathers (in which case there is a very slight possibility that the synagogue with the inscription might have been the “synagogue of the Freedmen” mentioned in Acts 6:9).

The Delphi inscription is a letter of around 52 AD from the Roman emperor Claudius. It was inscribed on stone at the Temple of Apollo at Delphi (above), although it now exists only as nine fragments. Of particular interest is this line (see also the photograph below):

[IOU]ΝΙΟΣ ΓΑΛΛΙΩΝ Ο Φ[ΙΛΟΣ] ΜΟΥ ΚΑ[Ι ΑΝΘΥ]ΠΑΤΟΣ …

Phonetically, that reads:

[Jou]nios Galliōn ‘o ph[ilos] mou ka[i anthu]patos …

This is a reference to Lucius Junius Gallio Annaeanus, who was briefly proconsul (*anthupatos*) of the Roman senatorial province of Achaea (southern Greece) at the time:

Junius Gallio, my friend and proconsul …

This same *anthupatos* Gallio appears in the New Testament (Acts 18:12–17: “Γαλλίωνος δὲ ἀνθυπάτου ὄντος τῆς Ἀχαΐας …”), and therefore provides a way of dating the events described there.

One of the fragments of the Delphi inscription, highlighting the name ΓΑΛΛΙΩΝ = Gallio (photo: Gérard)

I have written before about Papyrus Oxyrhynchus 29. It contains the statement of Proposition 5 of Book 2 of Euclid’s *Elements*, with an accompanying diagram (plus just a few letters of the last line of the preceding proposition). In modern Greek capitals, it reads:

ΕΑΝ ΕΥΘΕΙΑ ΓΡΑΜΜΗ

ΤΜΗΘΗ ΕΙΣ ΙΣΑ ΚΑΙ ΑΝ-

ΙΣΑ ΤΟ ΥΠΟ ΤΩΝ ΑΝΙ-

ΣΩΝ ΤΗΣ ΟΛΗΣ ΤΜΗΜ[ΑΤ]ΩΝ ΠΕΡΙΕΧΟΜΕΝΟΝ

ΟΡΘΟΓΩΝΙΟΝ ΜΕΤΑ Τ[Ο]Υ ΑΠΟ ΤΗΣ ΜΕΤΟΞΥ

ΤΩΝ ΤΟΜΩΝ ΤΕΤ[ΡΑ]ΓΩΝΟΥ ΙΣΟΝ ΕΣΤΙΝ

ΤΩ ΑΠΟ ΤΗΣ ΗΜΙΣΕΙ-

ΑΣ ΤΕΤΡΑΓΩΝΟΥ

However, the actual document (image below) uses “Ϲ” for the modern “Σ,” and “ω” for the modern “Ω”:

ΕΑΝ ΕΥΘΕΙΑ ΓΡΑΜΜΗ

ΤΜΗΘΗ ΕΙϹ ΙϹΑ ΚΑΙ ΑΝ-

ΙϹΑ ΤΟ ΥΠΟ ΤωΝ ΑΝΙ-

ϹωΝ ΤΗϹ ΟΛΗϹ ΤΜΗΜ[ΑΤ]ωΝ ΠΕΡΙΕΧΟΜΕΝΟΝ

ΟΡΘΟΓωΝΙΟΝ ΜΕΤΑ Τ[Ο]Υ ΑΠΟ ΤΗϹ ΜΕΤΟΞΥ

ΤωΝ ΤΟΜωΝ ΤΕΤ[ΡΑ]ΓωΝΟΥ ΙϹΟΝ ΕϹΤΙΝ

Τω ΑΠΟ ΤΗϹ ΗΜΙϹΕΙ-

ΑϹ ΤΕΤΡΑΓωΝΟΥ

This manuscript is important because, being from 75–125 AD, it dates to only four centuries after the original was written in 300 BC – most manuscripts of Euclid are twelve centuries or more after (in fact, it pre-dates the alterations made to the work by Theon of Alexandria in the 4^{th} century AD). The manuscript also contains one of the oldest extant Greek mathematical diagrams. The text is identical to the accepted Greek text, except for two spelling variations and one one grammatical error (τετραγώνου for τετραγώνῳ on the last line, perhaps as the result of the mental influence of the preceding word in the genitive):

ἐὰν εὐθεῖα γραμμὴ

τμηθῇ εἰς ἴσα καὶ ἄνισα,

τὸ ὑπὸ τῶν ἀνίσων τῆς ὅλης τμημάτων περιεχόμενον ὀρθογώνιον

μετὰ τοῦ ἀπὸ τῆς μεταξὺ τῶν τομῶν τετραγώνου

ἴσον ἐστὶ τῷ ἀπὸ τῆς ἡμισείας τετραγώνῳ.

It is really just a geometric way of expressing the equality (*x* + *y*)^{2} = *x*^{2} + 2*xy* + *y*^{2}, but in English it reads as follows:

If a straight line

be cut into equal and unequal [segments] (*x* + *y* + *x* and *y*),

the rectangle contained by the unequal segments of the whole (i.e. (*x* + *y* + *x*)*y* = 2*xy* + *y*^{2})

together with the square on the straight line between the points of section (+ *x*^{2})

is equal to the square on the half (= (*x* + *y*)^{2}).

The proof of the proposition is missing, however, and there are no labels on the diagram. I suspect that the manuscript was a teaching tool of some kind (either an *aide-mémoire* or an exam question). Alternatively, it may have been part of an illustrated index to the *Elements*.

Papyrus P52 is a small fragment written in a similar style to Papyrus Oxyrhynchus 29, but is dated a few decades later (to around 140 AD). In modern Greek capitals, it reads:

ΟΙ ΙΟΥΔΑΙ[ΟΙ]· ΗΜΕ[ΙΝ ΟΥΚ ΕΞΕΣΤΙΝ ΑΠΟΚΤΕΙΝΑΙ]

ΟΥΔΕΝΑ. ΙΝΑ Ο Λ[ΟΓΟΣ ΤΟΥ ΙΗΣΟΥ ΠΛΗΡΩΘΗ ΟΝ ΕΙ]–

ΠΕΝ ΣΗΜΑΙΝΩ[Ν ΠΟΙΩ ΘΑΝΑΤΩ ΗΜΕΛΛΕΝ ΑΠΟ]–

ΘΝΗΣΚΕΙΝ. ΙΣ[ΗΛΘΕΝ ΟΥΝ ΠΑΛΙΝ ΕΙΣ ΤΟ ΠΡΑΙΤΩ]–

ΡΙΟΝ Ο Π[ΙΛΑΤΟΣ ΚΑΙ ΕΦΩΝΗΣΕΝ ΤΟΝ ΙΗΣΟΥΝ]

ΚΑΙ ΕΙΠ[ΕΝ ΑΥΤΩ· ΣΥ ΕΙ O ΒΑΣΙΛΕΥΣ ΤΩΝ ΙΟΥ]–

[Δ]ΑΙΩN;

The reverse side also has writing:

[ΒΑΣΙΛΕΥΣ ΕΙΜΙ. ΕΓΩ ΕΙΣ TO]ΥΤΟ Γ[Ε]ΓΕΝΝΗΜΑΙ

[ΚΑΙ (ΕΙΣ ΤΟΥΤΟ) ΕΛΗΛΥΘΑ ΕΙΣ ΤΟΝ ΚΟ]ΣΜΟΝ, ΙΝΑ ΜΑΡΤΥ-

[ΡΗΣΩ ΤΗ ΑΛΗΘΕΙΑ· ΠΑΣ Ο ΩΝ] ΕΚ ΤΗΣ ΑΛΗΘΕI-

[ΑΣ ΑΚΟΥΕΙ ΜΟΥ ΤΗΣ ΦΩΝΗΣ]. ΛΕΓΕΙ ΑΥΤΩ

[Ο ΠΙΛΑΤΟΣ· ΤΙ ΕΣΤΙΝ ΑΛΗΘΕΙΑ; Κ]ΑΙ ΤΟΥΤΟ

[ΕΙΠΩΝ ΠΑΛΙΝ ΕΞΗΛΘΕΝ ΠΡΟΣ] ΤΟΥΣ Ι[ΟΥ]–

[ΔΑΙΟΥΣ ΚΑΙ ΛΕΓΕΙ ΑΥΤΟΙΣ· ΕΓΩ ΟΥΔ]ΕΜΙ[ΑΝ]

[ΕΥΡΙΣΚΩ ΕΝ ΑΥΤΩ ΑΙΤΙΑΝ].

Some clever detective work has identified the fragment as being from a manuscript of the New Testament gospel of John (John 18:31b–33 and 18:37b–38), permitting the reconstruction of the missing letters. The fragment is from the top inner corner of a book page (books with bound two-sided pages were a relatively new technology at the time, with many people still using scrolls). The fragment dates from less than a century after the gospel of John was written (and possibly just a few decades), thus helping in dating that work. There is no indication of any textual difference from later manuscripts – even the text on the missing parts of the front page seems of the right amount. The only exception is in the second line of the reverse side – there’s not quite enough room for the expected wording, and it seems likely that the duplicated words ΕΙΣ ΤΟΥΤΟ were not present.

In English, the passage reads:

… the Jews, “It is not lawful for us to put anyone to death.” This was to fulfil the word that Jesus had spoken to show by what kind of death he was going to die. So Pilate entered the Praetorium again and called Jesus and said to him, “Are you the King of the Jews?” …

… I am a king. For this purpose I was born and for this purpose I have come into the world – to bear witness to the truth. Everyone who is of the truth listens to my voice.” Pilate said to him, “What is truth?” After he had said this, he went back outside to the Jews and told them, “I find no guilt in him.”

The Akeptous inscription is one of a number of inscriptions found in the mosaic floor of a 3^{rd} century church which was discovered in 2005 while digging inside the Megiddo Prison in Israel (the date is just slightly later than the Dura-Europos church in Syria). The Akeptous inscription reads:

ΠΡΟϹΗΝΙΚΕΝ

ΑΚΕΠΤΟΥϹ,

Η ΦΙΛΟΘΕΟϹ,

ΤΗΝ ΤΡΑΠΕ-

ZΑΝ {Θω} {ΙΥ} {Χω}

ΜΝΗΜΟϹΥΝΟΝ

Phonetically:

Prosēniken Akeptous, ‘ē philotheos, tēn trapezan Th(e)ō Ι(ēso)u Ch(rist)ō mnēmosunon.

In English translation:

A gift of Akeptous, she who loves God, this table is for God Jesus Christ, a memorial.

Brief as it is, the inscription has several interesting features. First, Jesus Christ is being explicitly referred to as God, which tells us something about Christian beliefs of the time. Second, the inscription uses *nomina sacra* – divine names (“God,” “Jesus,” and “Christ”) are abbreviated with first and last letter, plus an overbar (this is denoted by curly brackets in the Greek text above). Third, the inscription records the gift of a prominent (presumably wealthy) female church member (the feminine definite article shows that Akeptous was female). And fourth, the reference to the construction of a table suggests that there were architectural features in the church to support the celebration of Communion, which tells us something about liturgy.

Our final inscription is a portion of the *Codex Sinaiticus*, a 4^{th}century manuscript of the Christian Bible, containing the earliest complete copy of the New Testament. This Bible is a century later than the Megiddo church, and two centuries after Papyrus P52. Unlike Papyrus P52, it is written on vellum made from animal skins, and is written in beautiful calligraphic script. I have selected the passage John 1:1–3a:

ΕΝ ΑΡΧΗ ΗΝ Ο ΛΟΓΟϹ,

ΚΑΙ Ο ΛΟΓΟϹ ΗΝ

ΠΡΟϹ ΤΟΝ {ΘΝ}, ΚΑΙ

{ΘϹ} ΗΝ Ο ΛΟΓΟϹ. ΟΥ-

ΤΟϹ ΗΝ ΕΝ ΑΡΧΗ

ΠΡΟϹ ΤΟΝ {ΘΝ}. ΠΑ[Ν]-

ΤΑ ΔΙ ΑΥΤΟΥ ΕΓΕΝΕ-

ΤΟ, ΚΑΙ ΧΩΡΙϹ ΑΥΤΟΥ

ΕΓΕΝΕΤΟ ΟΥΔΕΝ

In English:

In the beginning was the Logos, and the Logos was with God, and the Logos was God. He was in the beginning with God. All things through him were made, and apart from him was not one thing made …

In the Greek, *nomina sacra* for “God” can be seen, together with a number of corrections (including, on the last line, an expansion of the contraction ΟΥΔΕΝ = “nothing” to ΟΥΔΕ ΕΝ = “not one thing”). Spaces between words had still not been invented, nor had punctuation or lowercase letters, which means that it is almost impossible to make sense of the text unless it is read aloud (or at least subvocalised). Fortunately, things have changed in the last seventeen centuries!

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The Western Sydney University team reveals their new car (photo: Anthony Dekker)

Australia’s champion Challenger-class solar car team (not to detract from the two excellent Cruiser-class teams, Arrow and Sunswift) is Western Sydney University, who have just revealed their exciting new solar car (above and below).

Another view of the WSU car, which is a monohull design with a gallium arsenide array (photo: Anthony Dekker)

After being forced to trailer in the 2013 World Solar Challenge, Western Sydney University came 10^{th} in 2015 and 6^{th} in 2017 (see graph below). And that 2017 result did not do their car justice, because in 2018 they went on to defeat the second-place team, Michigan. Their superb new car (above) is engineered to be even faster. Could they win this year? Or does the Dutch team from Delft have a stranglehold on the top position? I guess we’ll just have to wait and see…

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The charts above and below (click to zoom) show the dimensions of some of the Challenger-class cars in the World Solar Challenge coming up this October (see also my illustrated teams list). In the chart above, ⬤ = cars with silicon arrays (4 m^{2} allowed), ⬛ = thin film single junction (3.56 m^{2} allowed), and ▲ = multijunction gallium arsenide (2.64 m^{2} allowed). All three technologies are in use this year. Hollow symbols denote cars from 2017.

Particularly noticeable is Twente’s incredibly shrinking car. They switched technologies this year, but were also so efficient that their new car is about 18% smaller than Delft’s – almost a square metre smaller! There are also three visible clusters – larger silicon-array cars at the top right, compact catamarans (like Twente and Delft) at the left, and monohulls at the bottom right. In the chart below, solid lines show dimensions for this year, and dotted lines those of 2017.

**Update**: the width of Eclipse’s entry has been corrected (the impact attenuator has been removed for WSC).

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The figures of Geometry and Arithmetic by the Coëtivy Master, late 15th century (detail from

For most of history, mathematics and the visual arts have been friends. Art was not distinguished from what we now call “craft,” and mathematics – geometry and arithmetic – provided both a source of inspiration and a set of tools. Polykleitos, for example, in the 5th century BC, outlined a set of “ideal” proportions for use in sculpture, based on the square root of two (1.414…). Some later artists used the golden ratio (1.618…) instead.

Symmetry has also been an important part of art, as in the Navajo rug below, as well as a topic of investigation for mathematicians.

The Renaissance saw the beginning of the modern idolisation of artists, with Giorgio Vasari’s *The Lives of the Most Excellent Painters, Sculptors, and Architects*. However, the friendship between mathematics and art became even closer. The theory of perspective was developed during 14th and 15th centuries, so that paintings of the time have one or more “vanishing points,” much like the photograph below.

Perspective in the

Along with the theory of perspective, there was in increasing interest in the mathematics of shape. In particular, the 13 solid shapes known as Archimedean polyhedra were rediscovered. Piero della Francesca rediscovered six, and other artists, such as Luca Pacioli rediscovered others (the last few were rediscovered by Johannes Kepler in the early 17th century). Perspective, polyhedra, and proportion also come together in the work of Albrecht Dürer. Illustrations of the Archimedean polyhedra by Leonardo da Vinci appear in Luca Pacioli’s book *De Divina Proportione*.

Illustration of a Cuboctahedron by Leonardo da Vinci for Luca Pacioli’s

Some modern artists have continued friendly relations with mathematics. The Dutch artist M. C. Escher (reminiscent of Dürer in some ways) sought inspirations in the diagrams of scientific publications, for example.

Today it is possible to follow in Escher’s footsteps by studying a Bachelor of Fine Arts / Bachelor of Science double degree at some institutions. There is also a renewed interest in the beauty of mathematical objects, whether three-dimensional (such as polyhedra) or two-dimensional (such as the Mandelbrot set). The role of the artist then becomes that of bringing out the beauty of the object through rendering, colouring, choice of materials, sculptural techniques, and the like.

Artistic techniques such as these (“must we call them “craft” or “graphic design”?) are also important in the field of data visualisation, and are recognised by the “Information is Beautiful” Awards. Speaking of which, this year’s awards are now open for submissions.

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The next calendar (**click for hi-res image**). Australian National Science Week is coming up on the 10^{th}. Why not get involved?

See more calendars here.

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