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Alan Turing, one of Britain's greatest mathematicians and thinkers, worked at Bletchley Park during the Second World War where he broke the Naval Enigma codes, allowing Britain to gather intelligence it needed to win the war. As well as developing plans for the Automatic Computing Engine, one of the first computers as early as 1945, he theorised on "the nature of spirit", showing "great interest in the paranormal phenomena of telepathy, clairvoyance, precognition and psychokinesis." Turing, who admitted to his onetime fiancée Joan Clarke that he had "homosexual tendencies", was arrested in 1952 for having a sexual relationship with a man and sentenced to a one-year course of female hormone treatment (used to “reduce the libido of sexual offenders”).

"Throughout his life, Turing broke the codes of science and society," the introduction panel tells us. This is the story of a man let down by his society, suffering a tragic fate as a result. However, the exhibition does not place undue focus on Turing's personal struggles, telling us just enough that we can be outraged. It focuses quite rightly on his achievements and thinking. There is an interesting sidestep to his boyhood, and relationship with a peer named Chris Morcom. Morcom can certainly be assumed to be Turing’s first love (and Morcom’s untimely death the reason Turing wrote and thought so extensively on “the nature of spirit”), but whether Morcom reciprocated Turing’s adoration isn’t clear. Morcom’s family have lent letters that Turing wrote to Mrs Morcom, Christopher’s mother, after he died. “Of course I simply worshipped the ground he trod on,” Turing writes.

It’s the code breaking theory and equipment that make the meat of this exhibition. Alongside the Pilot ACE is the part of the wreckage of a Comet Jet. When two of these, the world’s first airliners, crashed in succession, it spurred an urgent investigation hunting for a fundamental design flaw. Central to this investigation Turing’s Pilot ACE provided high-speed computation. Alongside this is a fascinating video of a real Comet imprisoned in an improvised water tank, which investigators used to replicate the pressure experienced by the jets at high altitude.

The curators display a selection of Enigma machines, Enigma working aids and differential analysers (mechanical calculators). Staggeringly, over 10,000 people worked at Bletchley Park by 1944. Its environment is painted as liberal, given the top secret nature of the work and therefore a level of camaraderie amongst staff. Nevertheless, this is where Turing met and proposed to mathematician Joan Clarke.

Other codes, other than man-made ones, also interested Turing. He delivered a pioneering paper on morphogenesis in 1952, exploring the link between mathematics and chemistry. Morphogenesis, which is the study of pattern formation in nature, is today generating considerable interest, despite having been largely abandoned after Turing’s death.

The exhibition’s objects, many of which seem linked to tragedy in some way, whether it be the tragedy of the war, or the tragedy of the Comet Jet crashes, are suitable mementos of Turing’s professional achievements. Two years after being sentenced to chemical castration (the other option having been a year imprisonment) Turing was found dead, the remains of cyanide solution nearby. The pathologist recorded a verdict of suicide, though Turing’s mother maintained his death had been an accident. The real tragedy, and the unspoken undertone of the exhibition, is that if he had lived longer (being just 41 when he died) there is no telling what other feats – scientific of philosophical – he might have accomplished.

Code breaker runs from 21 June to 31 July 2012 at the Science Museum