Key Plans and Tiles, a Self-Similar, Aperiodic Space-Planning System

Planning a floor from the furniture out, so that whatever is left over is always still worth leasing

Nothing in this paper constitutes an offer to sell, or a solicitation of an offer to buy, any interest in a Woodfine direct-hold solution; any offering is made only by the applicable Private Placement Memorandum. Statements marked "planned," "intended," "targeted," "may," or "expected" are forward-looking and subject to change. Full disclosures appear at the end of this paper.

Working Paper WCP-009 · v0.1.0 · CC BY 4.0

Key Plans and Tiles is a geometric self-similar aperiodic space planning system based on furniture and equipment arrangements and circulation. That is the formal definition, and the rest of this paper is an argument for why a developer would want one.

Our thesis is that a floor should be planned outward from the things that actually go in it — the desks, the examination tables, the laboratory benches, the paths people walk between them — rather than inward from an area figure divided by a headcount. And that once a floor is planned that way, the resulting layout should be allowed not to repeat. A floor built from one module repeated to fill the space can accommodate tenants whose needs are whole multiples of that module, and accommodates everyone else badly. A floor built from a catalogue of differently sized, geometrically complete pieces can accommodate a much wider range of tenants, and — this is the property that matters most — whatever is left over after each leasing decision is itself still one of the pieces.

Nothing described here has been tested against a built building, because no Woodfine Building has been built. The system exists as a design discipline and a prototype specification, and it is the discipline the prototype designs are being drawn against. Where this paper says something works, it means we have reasoned it through and drawn it, not that we have watched it lease.

1. The thesis

Planning from furniture and circulation first, in a system that never repeats by simple translation, is what lets a single floor plate serve tenants of very different sizes without wasted space and without awkward subdivision.

The claim is comparative, and the comparison is to a modular grid — the ordinary way a commercial floor is planned, in which the floor is divided into identical repeating units and tenancies are assembled from whole numbers of them. A modular grid is not a naive choice. It is easy to draw, easy to price, easy to explain to a tenant, and it makes the structural engineering straightforward. We are not arguing that it is incompetent. We are arguing that it has a specific and unavoidable failure mode, that the failure mode is expensive, and that it is avoidable by planning from a different starting point.

The failure mode is the remainder. A grid-planned floor leases well until the geometry stops cooperating, and then it produces a leftover: a strip too narrow, a corner too deep, a piece that is not a whole number of modules and is therefore not really a product. That leftover gets discounted, or amalgamated into a neighbouring tenancy at no rent, or sits. Our position is that a system planned from real accommodation, with pieces of genuinely different sizes that nest into one another, can be arranged so that the remainder after any leasing decision is always still one of the valid pieces. Not "usually." That is the design target, and it is the property the word self-similar is carrying.

2. The problem, in the reader's terms

Two kitchens make the point better than a floor plan would.

The first kitchen is built from a uniform grid of identical cabinet units. It is quick to specify and quick to install, and it works well until it meets something that is not a whole number of cabinets wide: a dishwasher, a chimney breast, an awkward corner by the door. Then there is a filler panel — a strip of matching material that closes the gap and does nothing. Every kitchen built this way has them. They are not a defect in the installation; they are what a uniform module does when it meets a world that is not a multiple of itself.

The second kitchen is planned around the appliances and the way the cook moves. The units come in several widths because the things that go in them come in several widths. There is no filler panel, because the plan started from the objects rather than from the module.

A commercial office floor is the same problem at a larger scale and with money attached. The conventional approach starts from an area figure — historically something in the region of a hundred and fifty square feet per person — multiplies it by a headcount, and produces a tenancy. That convention is under real strain, because it is an average that was never intended to describe any particular business and increasingly describes fewer of them: updated building codes, ventilation and wellness standards, and the actual spatial requirements of a dental surgery, a laboratory, a legal practice, and a software team diverge far more than an area-per-person figure can express. Planning the floor from the average produces the filler panels.

The industry has known for decades that the shape of the workplace follows the work rather than an area allowance [duffy-1997-the-new-office]. The proposal here is simply to take that seriously at the point where the building's geometry is fixed, rather than at the point where a tenant fits themselves into it.

3. How a commercial floor is actually divided

A reader who has never leased commercial space will benefit from knowing how the division ordinarily works, because the vocabulary is doing real work later.

A floor of a commercial office building has a core — the elevators, the stairs, the mechanical risers, the shared washrooms — and a perimeter of leasable space around or beside it. The leasable space is divided into tenancies by demising walls: the permanent partitions that mark where one tenant's premises end and the next begin. Demising walls are not furniture. They run floor to structure, they carry the fire separation between tenancies, and moving one is a construction project rather than a rearrangement.

Where those walls can go is therefore the single most consequential geometric decision in a commercial building, and it is usually made late and by default — inherited from the structural grid, which was itself set by whatever beam span was most economical. The result is that the column spacing chosen to save money during construction determines, for the next fifty years, which tenants the building can house.

Two further pieces of vocabulary matter. The structural grid is the arrangement of columns and beams holding the building up; the distance between columns is the bay, and the depth of a bay is how far back from the window face the leasable space runs before it reaches a column or the core. And what a tenant pays for is measured under published conventions — what counts as the tenant's own space, what counts as their share of the shared corridors and washrooms, and where exactly the line is drawn on a demising wall [boma-measurement-standards]. Those conventions decide how a given piece of geometry turns into rent.

4. Self-similar, and aperiodic

Both words in the title are precise, and one of them is easy to get wrong.

Self-similar means the same organising logic governs the layout at more than one scale. In this system there are three. The Key Plan is the smallest unit of space worth leasing — a room-scale plan sized from real furniture placement, real circulation, and real daylight access, with Key Plan sizes varying by what the tenant does rather than following one module. Key Plans combine into Tiles; a Tile is the block a tenant actually leases, and it is also the single zone the building's heating, cooling, and ventilation are sized to. Tiles combine, with the building's core, into the Floor Plate. A Key Plan nests into a Tile the same way a Tile nests into a Floor Plate, and — the load-bearing part — what is left over at any scale after a piece is taken is still a valid piece at that scale.

Aperiodic is the word most often mishandled in descriptions of this kind of system, and it is worth stating carefully, because the readers most likely to find this paper interesting are exactly the ones who would notice a loose version of it.

The clearest real-world example of aperiodic order is a Penrose tiling: a way of covering a plane, discovered in the mid-1970s, using a very small number of tile shapes — in the best-known version just two rhombi [penrose-1974-aesthetics] [gardner-1977-nonperiodic-tiling]. The tiles themselves recur constantly; the same two shapes are used everywhere. What never recurs is the pattern. Slide a copy of a Penrose tiling over itself in any direction, by any distance, and it will never line up — there is no translation that maps the pattern onto itself. That absence of translational repetition is what aperiodic means. It does not mean that no shape is ever used twice, and the tiling is emphatically not random: it is highly ordered, with the same local configurations appearing again and again, just never in a repeating lattice [grunbaum-shephard-1987-tilings-and-patterns].

That is the property being borrowed. A modular grid is defined by translational repetition: its whole character is that sliding it by one module reproduces it exactly. Key Plans and Tiles uses a small, defined catalogue of pieces — the same Key Plan types and Tile types throughout the portfolio — arranged in a sequence along any given floor that is set by the actual tenants on that floor and does not repeat. Same pieces, no repeating lattice. Order without periodicity.

5. Why this beats a well-optimised modular grid

A fair version of the opposing case has to be stated first, because a badly optimised grid is not the competition. A well-optimised modular grid, in a market where tenant sizes cluster tightly around a few values, is very hard to beat. It is cheaper to draw, cheaper to build, and more legible to everyone involved. If a developer knew in advance that their building would be occupied by tenants of two or three predictable sizes, a grid tuned to those sizes would be the right answer and this paper would be an expensive way to arrive at a worse one.

Our argument is that a developer building in smaller regional markets does not have that knowledge and should stop pretending otherwise. The tenants for a new office building in such a market are a genuinely mixed set: a law practice, a dental surgery, a medical imaging clinic, a regional government office, a small laboratory, a branch of a national firm, an accountant, a two-person consultancy. They are not multiples of one another. And crucially, a developer cannot know the mix before the building exists, because in these markets the building is the first of its kind and the demand it uncovers is not the demand that was visible before it was built.

From that starting point, four consequences follow, and they are the substance of the claim.

The remainder is always a product. This is the one that matters. If the leftover after every leasing decision is still a valid configuration, then the developer never holds an unlettable strip, and never has to discount a piece of floor into a neighbouring tenancy to get rid of it. It also means the building does not have to be leased in a particular order to lease well. A grid-planned floor has good and bad sequences of tenants; this one is meant not to.

Nothing is built until it is leased. Because the unbuilt remainder is always describable as some combination of the catalogue's pieces, the developer can leave it unbuilt. Only what has been leased gets constructed. Capital is not spent partitioning space speculatively, and the space that has not been let is not carrying the shape of a guess about who might take it. The corridor is built out on day one so that expansion is possible; the tenancies are not.

The structural grid follows the leasing geometry, not the other way round. Once the Tiles are known, the columns are placed around them, so that a Tile can always be separated out as a leasehold with no column standing in the middle of it. This inverts the ordinary sequence, in which the beam span is chosen for construction economy and the leasable geometry is whatever falls out. The ordinary sequence produces bay depths that certain tenants — the ones who know exactly how deep their layout needs to be and will not pay for more — simply will not take, leaving the landlord holding depth that cannot be let for the life of the lease. That loss is invisible at the moment the beam is chosen and permanent afterward.

The mechanical services zone is a leasing boundary, not a second boundary. Because the Tile is both the unit a tenant leases and the unit the heating and cooling are zoned to, the engineers are zoning against a line that already exists in the leasing plan rather than drawing a separate set of zones that then has to be reconciled with it. Two sets of boundaries that nearly agree is a recurring source of expensive coordination; one set of boundaries removes it.

The system is not uniform everywhere, and the exceptions are worth naming rather than hiding. At building corners and around elevator lobbies, a standard Tile does not fit, and smaller purpose-built configurations absorb the awkward geometry — the same job a filler panel does in the first kitchen, except that these are designed as complete pieces and are leasable and serviced in their own right. The smallest private offices carry no circulation zone of their own and open directly onto the building's shared corridor. And two of the building types in the programme — single-storey retail and light-industrial space — skip the Tile level entirely, with their Key Plans composing straight into a floor plate, because their tenants lease at a scale where the intermediate step earns nothing.

One further honest note on construction. The structural system that best serves this geometry across the markets in scope currently looks like a hybrid — web steel beams carrying wood floors with a concrete topping — rather than a pure steel, pure concrete, or all-timber building. That is a current direction arrived at with structural consultants, not a settled specification, and the geometry described here is what the structural choice has to serve rather than the other way round.

6. What this buys a tenant

The tenant-facing consequence is narrower than the developer-facing one and should be stated without inflation.

A tenant taking space planned this way is choosing from a defined set of geometrically complete configurations rather than negotiating a custom partition layout after the fact. Each configuration is drawn against published standards for daylight, for air, and for the circulation widths a workplace needs — a desk within reach of a window rather than a fixed distance from one, fresh air sized per occupant rather than per square foot, corridors dimensioned for the number of people who will use them [en-12464-1] [arbstaettv]. The intention is that a tenant is not buying a promise that the space is efficient, but a piece whose efficiency was the reason it exists at that size. Whether that holds in practice is something a built, occupied building will show and a drawing cannot.

What it does not buy them is a bespoke layout. The catalogue is deliberately restrictive. A tenant who wants something that is not in it is free to build it — the demising positions are fixed, and inside them a tenant may do as they like — but they are paying for it and managing it, and the building will not be redesigned around them. That is a real constraint and some tenants will not accept it.

There is also a consequence the tenant does not see and the market eventually does. A building sized from the accommodation it was designed to hold, with no allowance added for planning uncertainty, has less floor area to construct, to heat, to cool, and to clean than a building sized from an average. Less building for the same tenants is a durable advantage rather than a one-off saving, and it is the same geometry producing it.

What this changes for the reader

The change is that "efficient floor plate" becomes something a reader can interrogate rather than accept.

Every developer's material claims efficient design. The specific questions this system makes available are harder to answer in the abstract. Where can the demising walls go, and who decided — the leasing geometry, or the structural engineer's most economical span? What happens to the leftover space after a tenant takes a partial floor: is it still a product, or is it a discount? Is the mechanical zoning drawn on the same boundaries as the tenancies, or on its own? And was the floor sized from what tenants actually need, or from an area figure divided by a headcount?

Those questions are askable of any commercial building. A developer who has planned this way can answer them in specifics. A developer who has not will answer with a general claim about flexibility.

The trade-offs are real. A catalogue of differently sized pieces is more expensive to design than one module repeated, and that design cost is paid before a single building exists. Restricting tenants to a defined set is a genuine constraint that will cost some tenancies. Placing columns to suit the leasing geometry rather than the cheapest span costs money at construction, in exchange for rent over decades — a trade we believe in and cannot yet prove. And the whole argument rests on a claim about the remainder always being a valid piece, which has been designed for and not yet observed in a leased building.

An open invitation

The comparison at the centre of this paper is testable, and we have not tested it.

To architects and space-planning researchers: the honest open question is whether an aperiodic system genuinely outperforms a well-optimised modular grid across a wide range of tenant-size distributions, or only in the specific range a smaller regional market presents. That is a simulation anyone with a distribution of real tenant sizes could run, and the answer would be worth more to us than another year of our own drawing. We would particularly like to know where the crossover is — at what degree of tenant-size clustering the grid wins.

To researchers in aperiodic order and tiling theory: we are borrowing a property, not a proof. The question we cannot answer is whether the remainder property we want — that whatever is left after any admissible subdivision is itself admissible — can actually be guaranteed for a finite catalogue of pieces in a bounded rectangle, or whether it can only be approached. If that is a known result in either direction, we would rather learn it than discover it in a building.

To structural engineers: inverting the usual sequence, so that the column grid follows the leasing geometry, costs something, and we have estimated that cost rather than measured it across a real programme. What does it actually cost, and is there a configuration where the two objectives coincide rather than compete?

And to leasing agents and tenant representatives working in smaller regional markets: the premise underneath all of this is that tenant sizes in these markets are genuinely dispersed rather than clustered. If that is wrong — if the demand in these markets is more uniform than we believe — then a simpler system would serve better, and we would rather be told now.

Conclusion

Key Plans and Tiles is a self-similar, aperiodic space-planning system: the same nesting logic at every scale, a small catalogue of differently sized pieces, and a layout along any given floor that is set by the actual tenants rather than by a repeating module. Aperiodic here means what it means in the mathematics it is borrowed from — the pattern never repeats by simple translation, though its component pieces recur constantly — and a modular grid is the exact thing it is defined against. The reason to build this way is not elegance. It is that the leftover space after every leasing decision is meant to remain a product rather than becoming a discount, that nothing need be built until it is let, and that the columns end up where the tenancies want them instead of where the beam schedule wanted them. We have designed for those properties. We have not yet leased a building that demonstrates them.

References

Duffy, F. 1997. The New Office. Conran Octopus.

Gardner, M. 1977. Extraordinary nonperiodic tiling that enriches the theory of tiles. Scientific American 236(1): 110–121.

Grünbaum, B., and Shephard, G. C. 1987. Tilings and Patterns. W. H. Freeman.

Penrose, R. 1974. The role of aesthetics in pure and applied mathematical research. Bulletin of the Institute of Mathematics and its Applications 10: 266–271.

Building Owners and Managers Association International. BOMA floor measurement standards. https://www.boma.org/

European Committee for Standardization. EN 12464-1 — Light and lighting: lighting of work places, part 1, indoor work places. Issued by CEN; distributed through national standards bodies.

Federal Ministry of Justice (Germany). Verordnung über Arbeitsstätten (Arbeitsstättenverordnung — ArbStättV), §12 (Verkehrswege). https://www.gesetze-im-internet.de/arbst_ttv_2004/BJNR217910004.html

Contributors

Prepared by Woodfine Management Corp.

How this paper was produced

This paper is grounded in the operating and development work the preparing staff do themselves, and in what they have learned from the people they work with routinely: the graphic designers, web developers, and software developers and engineers who build the platform alongside them, and the architects and structural, building-services, and civil engineers they develop buildings with. No outside professional reviewed or approved this paper, and nothing in it is professional advice. It was drafted and edited with AI assistance under editorial direction, and the analysis and conclusions are Woodfine's own.

Disclosures

Woodfine Capital Projects Inc. ("Woodfine") is the author of record and is the developer and promoter of the buildings and the space-planning system described in this paper; Woodfine Management Corp. employs the staff and retains the consultants who prepare this material and conduct the design work it describes. Woodfine has a direct commercial interest in the approach argued for here. This work was funded internally; no external research funding was received. Nothing in this paper constitutes an offer to sell, or a solicitation of an offer to buy, any interest in a Woodfine direct-hold solution; any offering is made only by the applicable Private Placement Memorandum, which prospective investors should review with their own professional advisors. Some statements above describe planned or intended future work; language such as "planned," "intended," "targeted," "may," and "expected" marks this forward-looking content, which is subject to change and does not constitute a commitment regarding future performance. No Woodfine Building has been constructed as at the date of this paper, and no claim here about leasing behaviour, efficiency, or construction volume has been observed in a completed building.

Data and reproducibility

There is no dataset behind this paper, and the reader should know which of its claims are of which kind. The mathematical facts about aperiodic tilings — that a Penrose tiling reuses a small number of tile shapes, that what never repeats is the pattern rather than the shapes, and that the absence of translational repetition is what aperiodic means — are established results in a published literature that anyone can check, and we cite where. The regulatory standards the geometry is drawn against are published instruments; the German circulation ordinance is free to read, and the European lighting and measurement standards are sold by their issuing bodies. The commercial-property measurement conventions referenced are likewise published by their issuing association.

Everything else is our own design reasoning. The Key Plan catalogue, the Tile types, their dimensions, and the rules governing how they combine are specifications we are drawing, not measurements we have taken, and their central claimed property — that the remainder after any leasing decision is still a valid configuration — is a design target we have drawn for and not yet demonstrated in a leased building. The comparison with a modular grid at the heart of this paper has not been simulated or measured by us or by anyone else that we are aware of. The structural direction mentioned is a current working conclusion from consultation, not a settled specification. No independent party has reviewed this system, tested its claims, or confirmed them.

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