J.

clapclapclapclapclap

  "Eight point four seconds, at two metres fifty one centimetres."

  Jerboa scowled as his rival leapt from the mat. "I could do that from a standing jump," he said loudly. The rival tossed his head in a 'you don't worry me' kind of way and returned to his seat on a low wooden bench, towelling his neck and swigging a glucose drink.

  Mr Jambres checked something on the sheet pinned to his clipboard. "He's still some way off your record."

  "Who is he?" Veda watched the boy in the green nylon shorts. His hair was a mass of candyfloss curls and his legs were like twigs.

  Jerboa snatched up his own glucose drink, swilled it round his mouth then spat it forcefully into a bucket. "A total tosser," he said. "Jacko something."

  "Jargo," corrected Mr Jambres. "Jargo..... something." Once again he glanced at his clipboard. "Jaconet."

  Jerboa twisted his thin shoulders and adjusted the white athletics vest with orange and purple trim, the logo stitched in the same colours on the front and the rear-

  VJ

  - standing for Velcro Jumper. In this world, VJ DAY had a totally different meaning from its mainstream counterpart. In addition to the white VJ vest, Jerboa was wearing white shorts, white socks and grey trainers. He would have looked quite athletic, thought Veda, but for the orange velcro knee pads, green crash helmet, purple velcro hand grips and discreet JASOn tattooed in purple and green on his right shoulder.

  The curly-haired boy glared at Jerboa who eyeballed him back. "He thinks he's good," Jerboa was sneering. "The Human Fly. Pah. Who's the County Champion?" he crowed. "The Northern Intermediate Champion? Me!! That's who." He thumped the VJ on his chest. "I'll show him good." The heat of competition had forged from the mild-mannered if occasionally boring young boy a huge ego-monster.

  Veda had attempted to make a case against spending Saturday afternoon in a Ralgex-and-plimsoll-reeking gymnasium watching teenagers hurling themselves against a wall. Jambres had merely blinked and told her it was very important for her to spend VJ Day with them and that she would understand in due course.

  "Here I go." He slid the hand grips over his knuckles and checked the buckles on the backs of his knees. Then he crammed the helmet over his head and strode to the mark, a man on a mission, a person of purpose. The first of three. The best time and height would count as his final score.

  With one final glare at Jargo Jaconet, he set off,

  Pounded down the sprung wooden floor

  Hit the springboard with both trainered feet

  Crashed against the velcro wall

  Hung like a spider

  sprawled, limbs splayed out to distribute his

  e i

  t

  h

  w g

  evenly in the centre of the green velcro mat

  then, slow ly

  slo wly

  sl owly

  s l o w ly

 

  the fibres

  tore

  free.

  Jerboa whooped as he fell through the air, hands and feet flailing, crashed onto the mattress and bounced from his back to his feet in one practised move, pumping his fist and nodding to the judges who had busied around his immobile figure with tape measures and stopwatches.

  "Fourteen point two seconds," A gasp. A cheer. "Three metres ten," said the Judge.

  "Garggghhh!" snarled Jerboa, roughly rubbing his face with a towel. "Beat that! If you can! Loser!" Jargo Jaconet merely shook his curly mop and dusted his hands with talcum powder.

  Veda had seen enough. She tucked the loose-leaf binder under her arm and exchanged the din of competition for the peace of a bench beside the canal which oozed gently behind the sports hall.

  Jerboa had introduced his "opera on themes mathematical and metamathematical" with an unnecessarily lengthy lecture outlining

  Gödel's Theorem of Incompleteness:

  To every ω-consistent recursive class κ of formulae there correspond recursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg (κ) (where v is the free variable of r)

  Or

  All consistent axiomatic formulations of number theory include undecidable propositions.

  (Proposition VI in "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I" 1931)

  And

  Fermat's Last Theorem:-

  The equation

  an + bn = cn

  has solutions in positive integers a, b, c and n only when n = 2 (and then there are infinitely many triplets a, b, c which satisfy the equation); but there are no solutions for n>2. I have discovered a truly marvellous proof of this statement, which, unfortunately, this margin is too small to contain.

  And so we turn to

  The Knot Theory Garden

  A mathematical and metamathematical opera by Jerboa Jenneting

  Characters

  Giulio A mathematician (tenor)

  Romana A student of maths (soprano)

  Fabbrio A topologist (bass)

  Pippi A metalogician (contralto)

  Synopsis

  The opera begins with mathematician Giulio attempting to solve Fermat's Last Theorem. Giulio is approached by Fabbrio, who needs help in untying a nine dimensional knot in seven dimensional space. Unknown to Giulio, Fabbrio is attempting to ensnare his rival in a mass of equations. Each time a solution is presented, Fabbrio sidetracks Giulio until finally they plunge into Gödel's Theorem. Giulio is trapped in a Free Recursive Loop until he is rescued by his student Romana and friend Pippi who blast their way through with TNT (not TriNitroToluene, obviously, but Typographical Number Theory. Fabbrio escapes through a series of Diophantine Equations.

  Giulio and Romana pursue him across the Slopes of Stability until they confront and defeat him in the Seventh Elementary Catastrophe, the six-dimensional graph of parabolic umbilic. Romana plots their escape by drawing ideas from her research into concepts of transversality as outlined in Structural Stability and Morphogenesis (René Thom, then of the IHES at Bures-sur-Yvette, 1972), the work on chaos theory of Christopher Zeeman (then of Warwick University), her own developing thesis on number theory and non-Euclidean geometry and the breakthrough book Bifurcations on a Rectilinear Grid (Thomas Lüger, Professor of Mathematics at the University of Jena, 1986).

  (N.B. Jerboa Jenneting's arrangement of Bifurcations on a Rectilinear Grid for flute, harpsichord and two 'cellos will shortly be available on Compact Disc from PO Box 42, Jarrow).

  Together Giulio and Romana induce parabolic failure through an isomorphic recursion, destroy the graph and trap their rival forever in a self-replicating transition network. The happy couple return to their lab and sing blissfully that-

  "a system is ω-incomplete if all the strings in a pyramidal family are theorems, but the universally quantified summarising string is not a theorem."

  The opera falls into three parts-

  Morphogenesis

  Transversality

  The Slopes of Stability

  with the third ("The Slopes of Stability") containing the orchestral suite "The Seven Elementaries" for large orchestra, doubled brass and wordless chorus, each movement built around a solo instrument, by Jerboa Jenneting.

  The Seven Elementary Catastrophes are:

  The Fold Catastrophe (feat. solo bassoon)

  The Cusp Catastrophe (feat. 'cello and double bass duet)

  The Swallowtail Catastrophe (feat. cor anglais)

  The Butterfly Catastrophe (feat. solo piccolo)

  The Umbilic Catastrophes - hyperbolic, (feat. euphonium and French horn)

  elliptic and (feat. viola and tam tam)

  parabolic (feat. triangle and other percussion, wood block and tubular bells)

  "The triangle solo in the final movement is particularly noteworthy, and the moment when it is joined by distant bell chimes is one of the most moving in contemporary music
" (Recorded Music Notes 42)

  (N.B. The Seven Elementary Catastrophes will also be available shortly on compact disc in all good record retailers and from PO Box 42, Jarrow).

  Veda glanced at some of the compositional sketches. Jerboa's notes were scribbled against the text:

  interrupted cadence V-VI

  tonic I subdominant first inversion (VI)

  if B minor supertonic triad IIB

  submediant triad root position VI

  perfect cadence here

  mediant chord in c# minor discord of an

  augmented fifth

  Although the opera is designed to be "through-sung" or "durchgesang", there are moments where the characters sing what might pass as arias as well as moments where the composer clearly intends a "sprechgesang" effect. An example of the aria approach (Ex. 1) shows the composer's use of a more traditional operatic form for a period of emotional tenderness whilst the example of sprechgesang (Ex. 2) indicates a willingness to embrace new compositional techniques to illustrate mathematical concepts and Ex. 3, the heart of the conflict, shows a grasp of tonal colouring although the counterpoint is stretched to its limit. The basis of the work is an eight note theme in (B-E-D-E B-E-A-D, repeated and inverted, mirrored and reversed.

 

  Ex. 1 (From Act Three)

  GIULIO: Oh, Romana, light of my life, and fire of my loins, when the hysteresis is not discontinuous/

  ROMANA (agitato):

  Or a free recursive loop should trap us in the elliptical/

  GIULIO:

  Never fear, never fear. A system is omega-incomplete if all the strings in a pyramidal family are theorems...

  TOGETHER (accompanied by soaring strings):

  ... but the universally quantified summarising string is not a theorem.

  Ex. 2 (from Act Two)

  FABBRIO (speaking over a ground bass):

  You will fail, my friend. Hilbert's Tenth Problem with a Diophantine Equation,

  an equation in which a polynomial with fixed integral coefficients and exponents is set to 0. For example, a = 0 or 5x + 13y - 1 = 0

  GIULIO groans, lowering his bloodied head.

 

  Ex. 3 (from Act Three)

  ROMANA

  Recursive number theory shows for example that (vigoroso, trionfale, stringendo)

  Q(n) = Q(n-Q(n-l)) + Q(n-Q(n-2)) for n>2 and Q(1)= Q(2) = 1

  FABBRIO:

  Never never never will you defeat me, Romana! an = bn + cn

  ROMANA (ruthlessly inverting the dominant seventh):

  F(n) = n - M(F(n-1))

  M(n) = n - F(M(n-1)) } for n >0

  F(0) = 1, and M (0) =0

  FABBRIO:

  I have no more room! The graph crowds in, it pulls at my soul...... I am narrowly circumscribed into the margiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiin

  FABBRIO screams as he falls from the saddle point of the six-dimensional graph. He clutches at the axis, his fingers give way and he drops from the singularity into the void.

  Veda stared away from the collection of black dots scattered across the staves on the page. The gunmetal waters of the canal lapped softly against the stone sides and rusty-runged ladder. From inside the hall came a whoop and a cheer. It had suddenly turned cold.