As diplomat, organiser-in-chief and the living icon of German football, Franz Beckenbauer appears omnipresent at this World Cup. Except, that is, in the one place his legacy might be expected to be felt most keenly: the pitch.
Der Kaiser, it appears, has no 21st century successor. Indeed, few of the serious contenders even accommodated a sweeper; for all the advantages of fielding a visionary behind the defence to orchestrate play, this tournament represents the triumph of the back four.
In 2002, all the semi-finalists fielded a back three, supported by wing-backs. Advance four years and seven of the eight quarter-finalists are devotees of the four-man defence. The solitary exception is Ukraine, who switched to three centre-backs after conceding four goals against Spain. Goalkeeper Olexander Shovkovskiy was not beaten again until he encountered Italy in the last eight, but they remain very much the exceptions.
It was their 1974 World Cup-winning captain who once asserted that Germans can't play 4-4-2. Jurgen Klinsmann, however, does not possess a modern-day equivalent of Beckenbauer; his choice of formation may be dictated by a shortage of high-class central defenders. Nonetheless, accentuating the positive has contributed to Germany's unexpected popularity. Indeed, time was when a cynic could have sniped that Germany could not entertain; now, they are among the most attacking teams in the competition.
The search for Beckenbauer's heirs could extend to Australia and Mexico.
Guus Hiddink has brought flexibility to his tactics in international management but, for the second successive World Cup, his base has been provided by a back three. As the deepest of the Socceroos' trio, Lucas Neill's anticipation and powers of recovery provided a shield behind the rest of the defence. Ricardo Osorio proved a similarly classy operator at the heart of the Mexican back three but the defensive playmaker was captain Rafael Marquez.
From a starting position as the right-sided centre back, he had a tendency to stroll forward imperiously and dictate play. When goals became a priority, Ricardo La Volpe made his move into midfield a formal arrangement.
The Argentine-born Mexico manager, it seems safe to suggest, could be Claudio Ranieri's choice among the men in the dugouts; an inveterate tinkerer, seemingly with several different formations up his sleeve and an assortment of players capable of filling various roles. The aforementioned Osorio, for example, became an auxiliary right winger when Mexico chased an equaliser against Argentina.
It was possible to detect method in La Volpe's madness; Osorio revealed himself to be an excellent crosser. By shuffling his pack to give his gifted defenders a chance to influence play further forward, La Volpe - unwittingly - brought to mind a phrase the unlamented long-ball evangelist Charles Hughes used: the position of maximum opportunity. Hughes used it to justify direct football; for several managers in Germany, the aim was to manoeuvre their finest players into dangerous positions.
It is inevitable that national teams, especially from countries with smaller populations, have an uneven distribution of talent. For Hiddink, the objective was to find Harry Kewell and Tim Cahill space just behind Mark Viduka; as the search for goals against Japan and Croatia became increasingly desperate, their brief was redefined to a position just behind a front three. In each case, however, his changes were rewarded.
With a similar intention, the Czech Republic's Karel Bruckner paired his two most extravagant talents, Pavel Nedved and Tomas Rosicky, as attacking central midfielders. Versus the USA, it worked superbly; thereafter it was hampered, partly because the Czechs had an unfortunate tendency to be depleted by red cards.
Constructing a winning formula from 11 men has its difficulties; the frequency with which many managers have been left with a reduced contingent of players has brought some ingenious solutions. Marcello Lippi's ability to reorganise his Italian team in a 4-3-1-1 system, aided by the formidable defending of Fabio Cannavaro and Gennaro Gattuso's insatiable appetite for harrying, has been rewarded. Bruce Arena's USA arguably played their finest football with a makeshift 4-3-1 shape after two red cards.
Luiz Felipe Scolari's response to the dismissal of Costinha against Holland was novel, but inspired. He removed his solitary striker, Pauleta, and opted for 4-2-3-0. Luis Figo, Deco and Cristiano Ronaldo became a counter-attacking trio brimming with skill while, deprived of a direct opponent, the Dutch centre backs appeared confused who to mark.
Less likely to figure among the list of managerial masterstrokes was Tunisia's Roger Lemerre's reaction to the red card his forward Zied Jaziri received against the Ukraine; he did nothing. In a must-win match for the Tunisians, his 4-5-0 formation, surprising as it may sound, yielded no chances.
If conclusions can be drawn, it is that a lack of ambition and a lack of flexibility have been punished. A shortage of attacking intent - whether from England, Ecuador, Switzerland - has rarely brought progress in the knockout stages while Jose Pekerman's decision to replace Argentina's creative fulcrum Juan Roman Riquelme with defensive midfielder Esteban Cambiasso backfired. Elimination against the forward-looking Germans, albeit on penalties, duly followed.
Meanwhile, Marco van Basten's rigid adherence to the typically Dutch 4-3-3 often consigned an uncomfortable Dirk Kuyt to the wing. Making the left flank a mirror image of the right can be less important than playing to footballers' individual strengths. Argentina, for instance, frequently fielded one defensive full-back (whether Fabricio Coloccini or Nicolas Burdisso) on the right while the marauding Juan Pablo Sorin advanced down the left. It was the source of many of their chances.
Pekerman should be credited, too, for recognising the significance of the combination of artist and artisan at the heart of the midfield. For Argentina, Riquelme was the natural playmaker - and in his case, very much the fantasista - with Javier Mascherano the self-sacrificial 'water carrier' behind him.
Like his Italian counterpart Andrea Pirlo - whose excellence was such that Francesco Totti's slow start to the tournament did not impede progress to the semi-finals - Mascherano provided proof that passing and positioning, rather than a willingness to snap at heels, are the essentials of an anchor midfielder's job (indeed, Costinha apart, it is hard to think of holding midfielders who have been dismissed).
Deco will resume the pivotal role in the Portugal team against France, who boast the greatest flair player of his generation. Zinedine Zidane appears to have rediscovered his joie de vivre in the hole behind lone striker Thierry Henry. Against Spain and Brazil, everything has flown through Zidane, courtesy of his unselfish adjutants Patrick Vieira and Claude Makelele.
The lines of demarcation are less strictly drawn in the German team, such is Michael Ballack's willingness to do the dirty work, but Zidane enjoys a truly free role; a chance to find and exploit positions of maximum opportunity for the French. For too long, however, Brazil failed to grant Ronaldinho the same licence, trying to shoehorn too many supposed superstars into one team and then shackling them with defensive duties, while the tiring Emerson failed to fulfil his brief as the supplier to the creator.
It has become a footballing cliché that 'players win matches'. So, at times, do managers and even systems. Fundamental to all three, however, is finding places where gifted footballers can exert an influence. In the 2006 World Cup, those two positions have been just in front of the back four and ahead of the rest of the midfield, but a fraction behind the forward line. Their occupants may meet with a regal nod of approval, for Franz Beckenbauer understood about finding and using space; he just did it in a different area of the pitch.