Translating Between Hexagon and Triangle

In addition to Numenta’s orientation, we may have identified or be close to identifying a suitable reference point of the grid cell lattice allowing us to frame our material in accordance with a Numenta standard.

Both hexagon and triangle tiling may be fractal-like and both seem to scale well in terms of reference frame insofar as both may be used to represent anything from a sub-atomic particle to a cosmos including a single brain cell, ensemble of neural activity, higher level functions or the organism and organisms performing such functions, so by way of example; if in addition to the triangle, a single hexagon tile of the Numenta grid cell matrix may correspond with a single neuron, then a tri-coloured triangle such as in (figs. 3a or 3c), may correspond with a single hexagon tile in which case the hexagon goes without being shown, otherwise the first hexagon appears at three units as in (fig. 3f) or at six units as in (figs. 3g and i) and within the surely remarkable shows in ([figs. 5a, 7 and 8).

(Both eight and twelve triangle configurations are currently unreleased and given a number of interesting possibilities in both cases, further experiments may be required to determine a correct arrangement at which point the floor with the triangle tiles may be open to suggestions.)

As in equal sized bubbles forming a sheet:

the geometry of what is being modeled is inherently hexagonal, and note that the waveform from this billion euro model is round, as though hexagon was further sub-divided.

It is not necessary to round out the signals, but I successfully went from 6 radials to 12, 24, then 48 or more in which case it’s not worth the extra processing time for such a small gain in roundness detail.

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Wow!! Didn’t see the fluid flow before; if we’re interpreting correctly it’s brilliant and may be dealing with something very difficult to describe otherwise, in addition may help resolve a matter of orientation within the triangle model. (hope we might come back to this important matter soon or as time may permit)

To be clear about triangle and hexagon; although coming from a different perspective and describing the matter in different terms and figures, we’re sure to compliment rather than compete with the hex tile schema as may be seen in the framework for the hexagonal pattern and how it relates to the triangle.

Regarding the dynamic being shown by what appears to be fluid-flow between the tiles, was that intentionally incorporated into the model or was it a pleasant surprise such as the video may suggest by the expression of delight? (nothing pressing; looking forward to hearing more and we agree; beautiful! Once again, bravo on quite a remarkable demonstration!)

If under the spell of Moser 2005 we’ve been hexed, triangle and rhomboid may prove a haunting experience.

Regardless, we’ve rendered (fig. 9) featuring gmirey’s lattice with at least one possible Numenta orientation where some options and their implications present themselves.

Strictly speaking, such unit scaling varies given the context, but in order to readily apply and translate between grid, triangle, hexagon and the physiology be it cell, ensemble or function; we may well need consensus on a standard unit. We suspect the answer may be unclear at the moment as we’ve been unable to correlate the necessary data points within the Numenta literature, but Bitking may be honing in on an SU using a measurement based method, which may play an important role going forward.

Figure 9

Just watched the 2012 Blue Brain Project video; great call, Gary! The video holds up well, offers among other notables a great depiction of the cortical column, a splendid soundtrack and couldn’t have hoped for a better companion piece to “thus spake”.

There may have been enough occurrence of Jungian style synchronicity along the way that one may become somewhat accustomed to the experience, but from our perspective this one may be a bit too freaky since where enough room for some lingering doubt remained previously, may now become a challenge to dismiss.

In order to facilitate translation between triangle and hexagon and to further our understanding of human and AI cognition such as in (fig. 7), we prepared a sample representation with parentheses added, based on what we could interpret from a text excerpt of Matt Taylor’s description in HTM School - Episode 13. As a base unit, (fig. 10) may offer no less than ten points of reference providing more than sufficient opportunity to correspond between triangle and hexagon in addition to answering a question of how columns might be shown. Using Matt’s description, the figure has been working for us so far and we’re hoping others may also derive some benefit.

Figure 10Geometry%20of%20Consciousness%20Label%20Sample

Example Key:
A cortical column gets feed-forward input (a), which comes from the direction of the senses (c ). Let’s say that a layer in the cortical column (1), is running those algorithms; the layer would tap into the cortical columns’ feed-forward stream (4), to get proximal input (5). The output of a layer can project to the inputs of other layers (5b), including itself (6c).