The invisible staircase is not a straight path, but twists unpredictably as it ascends. Although each segment connects, the degree of twisting far exceeds Shard's imagination. They move quite slowly, with the witch behind them tugging at the hem of Shard's shirt, watching closely where his feet tread.
When Shard tried to recreate in his mind the exterior appearance of the entire twisted staircase, the spirals and turns that defy normal human logic made him grunt and nearly stumble. The construction of this staircase employed terrifying knowledge, and its design is itself part of the knowledge.
The height of the tower's first floor is much taller than it appears from the outside. The three of them climbed at least twice the height visible from outside, yet still did not see the second floor.