Phyllotaxis – the arrangement of plant organs around their supporting axes – is an important attribute of plant architecture. Recent experimental data show that the first phase of typical phyllotactic pattern development in flower heads can be abstracted as an extension of the Fibonacci substitution L → LS, S → L. We demonstrate that the most intriguing geometric feature of phyllotaxis, the golden angle between the emerging organs, is a mathematical consequence of this substitution process.