I am an applied mathematician, interested in 'real world' phenomena where geometry is the driving force, while (low-dimensional) topology keeps an eye on things. I first researched these ideas in my PhD at Applied Maths at the Australian National University, supervised by Prof. Stephen Hyde and Dr Vanessa Robins. We looked at embeddings of graphs in space, which generalises knot theory as the branching allows tangled motifs which cannot be present in the knots and links of typical knot theory. Subsequently I did a Postdoc with Prof. Osamu Terasaki in the Dept of Chemistry (MMK) at Stockholm University in Sweden. There we looked at geometrically mediated self-assembly of mesoporous silica, which forms into delightfully complex yet periodically repeating shapes, on a length scale much longer than the constituent molecules. Following this I did a second Postdoc with Prof. Randy Kamien in Physics at UPenn, USA. I mainly looked at kirigami, which is the art and science of cutting, rejoining and folding a flat sheet into a desired 3D configuration. The cutting and rejoining create intrinsic curvature, while folding generates extrinsic curvature. Combining both methods allows the creation of a wide variety of shapes. Kirigami is widely useful because the underlying principles are (modulo some engineering concerns) applicable at different sizes from atomically thin sheets such as graphene up to buildings and large structures to be deployed in space which need to first fit inside a rocket's cargo bay.

I am a keen rock climber, horse rider, and sailor, when I have access to big rocks, horses or boats. That's me in the photo above, belaying a climber beneath me at Point Perpendicular on the NSW South Coast.


Below are my publications in reverse chronological order, as of January 2018. It is not my intention to actively maintain this page, so check with your preferred academic search engine for more recent information. If you can't get access to one of my papers, let me know and I'll try to help you out. (It's not hard to find my email address, I'm the only Toen Castle.)

UPenn postdoc, and legacy work

12. Optimal packings of three-arm star polyphiles: from tricontinuous to quasi-uniformly striped bicontinuous forms
L de Campo, T Castle, ST Hyde; Interface Focus 7 (4), 20160130

11. Additive lattice kirigami
T Castle, DM Sussman, M Tanis, RD Kamien; Science advances 2 (9), e1601258

10. Algorithmic lattice kirigami: A route to pluripotent materials
DM Sussman, Y Cho, T Castle, X Gong, E Jung, S Yang, RD Kamien; Proceedings of the National Academy of Sciences 112 (24), 7449-7453

9. Making the cut: Lattice kirigami rules
T Castle, Y Cho, X Gong, E Jung, DM Sussman, S Yang, RD Kamien; Physical review letters 113 (24), 245502

Stockholm postdoc, and legacy work

8. The role of curvature in silica mesoporous crystals
Keiichi Miyasaka, Alfonso Garcia Bennett, Lu Han, Yu Han, Changhong Xiao, Nobuhisa Fujita, Toen Castle, Yasuhiro Sakamoto, Shunai Che, Osamu Terasaki; Interface focus 2 (5), 634-644

7. Trading spaces: building three-dimensional nets from two-dimensional tilings
T Castle, ME Evans, ST Hyde, S Ramsden, V Robins; Interface Focus 2 (5), 555-566

6. Entanglement of embedded graphs
T Castle, ME Evans, ST Hyde; Progress of Theoretical Physics Supplement 191, 235-244

5. Shape- and Size-Controlled Synthesis in Hard Templates: Sophisticated Chemical Reduction for Mesoporous Monocrystalline Platinum Nanoparticles
Hongjing Wang, Hu Young Jeong, Masataka Imura, Liang Wang, Logudurai Radhakrishnan, Nobuhisa Fujita, Toen Castle, Osamu Terasaki, and Yusuke Yamauchi; Journal of the American Chemical Society 133 (37), 14526-14529

4. Bicontinuous Cubic Mesoporous Materials with Biphasic Structures
AE Garcia-Bennett, C Xiao, C Zhou, T Castle, K Miyasaka, O Terasaki; Chemistry-A European Journal 17 (48), 13510-13516

PhD publications

3. Entangled graphs on surfaces in space
My PhD thesis, done at the Australian National University under the supervision of Stephen Hyde and Vanessa Robins in the Dept. Applied Maths, Research School of Physics.

2. All toroidal embeddings of polyhedral graphs in 3-space are chiral.
T Castle, ME Evans, ST Hyde; New Journal of Chemistry 33 (10), 2107-2113

1. Ravels: knot-free but not free. Novel entanglements of graphs in 3-space.
T Castle, ME Evans, ST Hyde; New Journal of Chemistry 32 (9), 1484-1492