What the geometry is, and what to do with it.

What the golden ratio is, why 1.618 shows up in shells, sunflowers and paintings, and how to see it in a work of art. A plain explanation with examples.

Where to hang a picture, how big it should be and how to space a group — using the golden ratio as the rule of thumb. Measurements in centimetres, no guesswork.

How to choose, size, license and produce large-format wall art for a spa, clinic, hotel or office — what the commercial licence covers, which surfaces last, and what it costs.

How to turn a high-resolution art file into a print you are proud of — which size the pixels allow, which paper and finish, where to print, and how to frame it.

Why Gustav Klimt’s gold period still looks modern, how the spiral and the golden section run through his patterns, and how the Klimt Reimagined collection rebuilds them on the Fibonacci sequence.

What Leonardo actually did with the divine proportion, why his notebook pages look the way they do, and how the Da Vinci Reimagined collection rebuilds them on the Fibonacci sequence.

Where the golden ratio really appears in buildings — the Parthenon claim, the Great Pyramid, Notre-Dame, Le Corbusier’s Modulor — what is myth, what is measured, and how to use 1.618 on a wall today.

The nautilus is the poster child of the golden ratio — and it is not quite a golden spiral. What the shell actually does, how far off it is, where the ratio really lives in nature, and why I draw the shell the way it would be if it obeyed.

What the Fibonacci sequence is, where it comes from, how it turns into the golden ratio and the golden spiral, and how to draw the construction yourself in five minutes — with the sizes for a print.

Sofa, bed, dining table, desk, hallway, bathroom, stairs: the right picture width for each, worked out from one rule — furniture width divided by 1.618 — with standard print sizes and pairs and triptychs included.

How to actually use the golden ratio in a photograph — the phi grid, the spiral overlay, where the subject and the horizon go — and when the rule of thirds is the better tool. Plain, with numbers.

Apple, Twitter, Pepsi, National Geographic: which logos really use the golden ratio, which claims are drawn on afterwards, and how designers actually use 1.618 — in type scales, layouts and spacing — without the mysticism.

Sunflowers, pine cones, nautilus shells, hurricanes, galaxies, the human body, DNA: a sober list of where the golden ratio and Fibonacci numbers really appear in nature, where they only nearly do, and where the claim is wishful measuring.

Type a width and get the golden-ratio answers: the right picture size for a sofa or wall, the golden section point, the height of a golden rectangle, the Fibonacci unit and the nearest standard print size. Runs in your browser.