I did have a look at your web-site, though I think I have only read the surface material, and the idea of breaking away from the Jekyllian mould of your mother and the Pope’s is an exciting and interesting one. As regards the design itself however, I find it very difficult to comment on other people’s work at this early stage, and in the case of the parabola I don’t feel I have sufficient grip on what is proposed - what you expect it to be like as an experience, what the effect of the planting will be, and so on.
What I can do is list the kind of questions I would ask, and that you have probably already asked, of the design.
1 There are views from the garden up to the surrounding ridge and (potentially) out into the parkland from the slip on that side. The first two questions that occur to me then are: is the design to be inward-looking (are the views out to be significant) and second are the views in important (is the whole geometry of the design to be seen from one or more points, and if so where exactly are these points, and what will be done to ensure that the garden is seen from them and only from them).
If the geometry is not perceptible either from within the garden or from outside it, and if it has no cultural resonance (it appears to be a purely mathematical construct), then I think one would have to wonder if it was worthwhile. In terms of relating to the outside, I would say that the most exciting approach would be to reopen the yew walk from the house, and to link the garden and walk into your library, and the two obvious view-points are the Settle, and the high ground east of the garden (particularly if such a view-point could be added to the Marl Pits loop).
2 If the geometry is to be seen in one go from outside the garden, then it will very much affect the height, style and massing of the planting, and I think that will have to be addressed in outline at an early stage - I get the idea of a lattice of paths holding down bulbous beds, rather like a geodesic design, but it could be that the designers want the planting to cut right across their geometry and pay no respect to it.
3 The geometry of walled gardens usually has a focal point (the junction of the two cross-paths); it is easy to think of other walled gardens that are polyfocal, with a series of spaces each with its own centre. This geometry has no focal point, it is what I would call a tessellated design. It has been tried in gardening, and it has had great successes (e.g. the ferme ornée, the idea of ‘garden rooms’), but it is not easy to bring off. If the walls at Hadspen really have the line of a parabola, then it will have a focal point (any line radiating from this point will ‘bounce’ off the curving walls in a line that is parallel to every other line; presumably all these lines will cross the straight arm of the D at a right angle).
4 It also has no cultural meaning (culture can be of any kind, paths laid out to get from A - B, to give easy access to every part of the garden, to conform to heaven-knows-what religious/scientific symbolism). This design is stripped of association, the paths are not tied in to the doors, and some of the access is decidedly circuitous. It starts therefore from profoundly un-English premises (English design tending to be empirically-based). This is probably what attracted you to it.
5 One shortcoming of tessellation is that the spaces that it generates tend to be similar. In this case there are some bigger beds, but these are so big that they may have to be broken up with hidden paths simply to allow them to be managed. In gardening terms there are no long vistas, no glades, no emptiness, no seats, but a maze-like series of paths each of which is the same width and of similar character to the next (even the two zig-zag sections appear to have the same dimensions). There will be other gardening challenges - for example it is always difficult to plant the acute angles of a bed which is surrounded by paths - people, particularly people with horticultural tractors, will tend to walk and drive across the corners. None of this makes the garden impossible, just a challenge.
As it happens I have written two articles about geometry and one (in fact the second of the two though for some reason it is to be published first) comes out in Garden History at the end of this month, I shall send you a copy. It might be interesting to put it up against your design.