Terracing with Spectres

Maja Kuzmanović & Nik Gaffney

2024–2026

The making of an aperiodically tiled stone terrace for FoAM’s Istrian studio, tiling with a shape that was only proven possible months earlier.

In mathematics, a single shape that tiles a plane without ever settling into a repeating pattern is called an aperiodic monotile, or an einstein, from the German ein Stein: one stone. The Spectre is such a shape, identified in 2023, at the overlooked midpoint in a family of forms its discoverers had previously published. Unlike its predecessor, the hat, the Spectre tiles with no mirror images, only rotations. Spectres cast no reflection.

October 2023. Stone cutting template with dimensions and numbers of tiles needed to cover the terrace.

Ein Stein, one stone. A chiral aperiodic monotile (2024). On aperiodic tiling as a form of pattern thinking, see Pattern Thinking Across Domains. Here, that one stone became 480, CNC-cut from Istrian limestone. A property that was elegant in the abstract – no flipping – becomes a material constraint.

While renovating FoAM’s new studio in Istria, we were looking for interesting ways to resurface a terrace using local limestone. Discussing possibilities with Kamen Pazin, who quarry and cut stone, they mentioned their CNC waterjet, able to cut stone into any shape. Having read about the Spectre monotile, we were keen to try – if we could find a stonemason willing to work with the tiling. Davor Baf accepted the challenge. When we showed Davor the sketch, he expected it would be “a simple matter of rotating each piece until it fit”. This assumption led to a months-long exercise in ad-hoc combinatorics.

What follows is reconstructed from Signal messages exchanged between Brussels and Istria, June–September 2024: templates corrected overnight, photos of deliveries and small disasters, unanswered voice calls, and the slow accumulation of stone on concrete.

20 June 2024

MK: They’re here! The Spectres have arrived 🏗️

MK: Delivery done. All 480 pieces present and accounted for. Sanded and brushed on one side. They look and feel great. Now waiting for the stonemason’s reply...

3 July 2024

[Unanswered voice call]

MK: Davor, the stonemason, arrived and started ‘the puzzle’. I gave him the template we printed in Brussels, but it looks like something is wrong. Is this the latest version? It's on the last page of the "aperiodic-terracing-notes.pdf" file on neon. It looks like quite a few tiles have to be turned upside down to make the pattern on this template. I thought we didn’t have to do this using the spectres?

NG: in the printout i thought the dark tiles were focal points, rather than reflections. so yes, it's a different tiling.

MK: The tiles are sanded and brushed on one side only. The ‘bottom’ of the tiles is unfinished and looks weird (not in a good way). What do we do?

The first template was generated from a tiling that included reflected (mirror-image) tiles – a hat, not a spectre. Since the tiles were sanded and finished on one side only, reflected tiles would need to be laid face-down.

[Incoming voice call]

NG: this one should work. i thought i was making it "easier" but obviously extra complications 🤪

MK: First row... IT WORKS. There was a bit of panic here I have to admit.

NG: would you agree that this size & position agree with the photos?

MK: the rightmost tile's left corner is cut by about 3 cm. The floor plan doesn’t match the built terrace exactly 😬 The edge of the terrace is more like the red line.

NG: hmm. just need to expand it a bit now... also can you check the 30cm vs 40cm calculations (tiles should be 30cm).

MK: the squares in which the tile fits should be 38 x 30.

NG: it’s a bit too much of a challenge at the moment to find our layout in this...

MK: let’s just go to sleep.

NG: its there somewhere!

4 July 2024

NG: realised last night that it's not that easy to expand the region without changing other parts of the layout. i assume the first section is now glued to the terrace and can't be moved.

MK: Yes. Should we stop tiling? Maybe a quick voice chat?

[Outgoing voice call]

NG: getting systematic. narrowed down the search space, but that small rectangle is the terrace. i think i've found something. but still need to check... are you there now?

MK: I am. Here’s the overhead photo of the in-progress tiles, with the template of the sections that are correct. so far so good! Looks like a match, look the same to you?

NG: yes! A bit further down there is a single imperfection (nod to the Girih tilers?) so would need to cut one tile to shape. otherwise fits

MK: can you double check that the edges under the blinds are ok?

NG: ok. here’s the expanded layout with overflow. NOW it should work

MK: It does work, yay! But... we really need a template with the focal points in different color. It’s tricky to follow the pattern on this.... X marks the tiles that are placed and glued.

[Outgoing voice call]

5 July 2024

NG: new template with some coloured tiles

MK: New template Printed and Ruggedised. This works much better. Aand.... we got to the one imperfect tile! Cute ;) No problem to cut one of the broken pieces in this shape (Davor will do it at the end).

MK: Six hours later...

22 August 2024

Two months, a surgery, 3000km and a long hot summer later...

MK: Pattern complete, but...

MK: It doesn’t quite reach the edge.

[Outgoing voice call]

NG: terrace-SW-PR.spectre.extended.pdf

5 September 2024

MK: Packing away the paper template!!! Just the edges remaining – they’re better cut by measuring the actual holes than following the drawing.

Where the template reaches its limit, the hand takes over. On the gap between instruction and resistance, see Crafting at the Threshold of Knowing.

[Incoming voice call]

MK: Ugh... I haven’t thought of this when agreeing to bury the downpipe instead of letting the water run over the terrace... further tweaking required.

13 September 2024

[Unanswered voice call]

MK: Davor isn’t happy: the floor isn’t flat enough – there are these tiny ‘teeth’ on some junctions, because the concrete beneath the tiles wasn’t properly flattened and levelled [insert copious sighing and swearing at the builders]. Suggestion: resand and re-brush the whole floor. Davor has to test different types of sand and brushes, to match the surface of the tiles we received in June. I’m fine with that even if it means further delays (it’s a bit dangerous now). One thing that’s annoying though – if we knew this, we could have used the first template with reflected tiles all along!

[Incoming voice call]

17 September 2024

MK: Sanded and brushed, washed by the rain. Now just the grouting...

19 September 2024

MK: Here it is, grouted! Blue-grey grout brings out the pattern quite well. Aaand – spot the ‘imperfection’ ;)

[Incoming voice call]

MK: Done.

Phew! Just in time...

With gratitude to Davor Baf, Marijan Pulić, Kamen Pazin and everyone else involved, for their patience, skill and creativity.