Something strange is moving slowly through the august rooms of London’s Royal Academy of Arts. Something strange and slow and red and leaving a sticky wax trail behind it. Elsewhere, great gobbets of the same stuff are being fired from a cannon, shooting at 50mph from one room to another, through the grand wooden doorway that separates the Large Weston Room from the smaller gallery of the same name.

Boom! Splat! At 20-minute intervals the normally stately grandeur of the Grade I listed building is disturbed by cannon shot and a wax missile hitting the wall – and normally unruffled art goers jump or giggle or look at one another questioningly. Meantime, a heap of wax slowly accumulates in the corner of the room while splatters of blood-red wax dot formerly pristine white walls.

Shooting into the Corner (2009) is an artwork that will grow and take on its own form until this extraordinary exhibition of Anish Kapoor’s sculptures ends its run on December 11.

This is the first time the RA has held a solo exhibition of a living British artist in its main galleries. It’s not a retrospective so much as a survey of the work of the Mumbai-born artist as it includes new experimental pieces. Kapoor, one of the most influential and pioneering sculptors of his generation, won the Turner Prize in 1991, was elected Royal Academician in 1999, and was awarded a CBE in 2003.

The first piece I mentioned, the monumental train-like Svayambh (a Sanskrit word roughly translating as ‘auto-generated’) moves almost imperceptibly along a track that runs through five galleries (as pictured). Three of those galleries are blocked to view. You see the artwork at only three stages of its hour-and-a-half transit, watching as the great block of red wax squeezes its way through doorways, smearing all it touches. The work forms the “backbone” of a very varied, imaginative exhibition, and was previously shown in Nantes and Munich (2007). It has various connotations dependent on its location. According to RA curators, it conveys uncertainty, destiny, and since it moves at glacial pace, “inexorable un-human time”. It expresses Kapoor’s interest in “self-authored sculpture” – hence its name.

Also creating themselves, this time via a pre-set programme on some sort of concrete extruder in his south London studio, were the heaps of grey concrete, looking like different sized giant slugs, worm-casts, or turds, that make up the field of work that paradoxically goes under the poetic title Greyman Cries, Shaman Dies, Billowing Smoke, Beauty Evoked (2008-9). It’s a bold experimental work and one that the artist, whose more typical works are often celebrated for offering profound spiritual engagement, doesn’t explain. He opts out of this, saying it has an interesting relationship with the works around it (which it does, though I found them rather repellent), that he’s interested in “process” and prefers to “leave forms to speak for themselves”. I couldn’t decide what to make of these archaic or organic (?) extrusions. So, I just registered them as evidence of fresh experimentation from a highly original artist and moved on.

But outside, before you get to these – and, fear not, there are some of his more familiar styles here too, among them pigment pieces from the late 1970s, geometric shapes covered in beyond intense primary colours; his stainless steel reflective sculptures, those concave mirrored “non-objects” at once beautiful and fun to walk around, each piece fully involving its surroundings; and an utterly fabulous indented wall of chrome yellow so meditative you lose yourself in it – you have to walk through the Annenberg Courtyard. Here is a new work made for the show: Tall Tree and the Eye – an apparently random tower of 75 polished steel baubles reflecting themselves and surrounding Palladian architecture, giving back to the viewer a glorious multiplicity of images of courtyard, people and passing clouds. Seemingly weightless and unfixed, you wonder why they don’t just float like balloons into the sky above.

Seen from Piccadilly, there could be no better advert for an ingenious show full of contrasts and tensions, one that works at spiritual and visceral levels.