Complex Projective 4-Space

Moritz Firsching asked in 2016 whether there exists a dodecahedron, combinatorially equivalent to a regular dodecahedron, with rational vertices lying on the unit sphere. The difficulty arises from the combination of three constraints:

1. The twelve pentagonal faces must all be planar;
2. The vertices must all be rational points;
3. The vertices must all lie on the unit sphere.

Conditions 1 and 2 are realised by Schulz’s dodecahedron (shown in Firsching’s question). Conditions 1 and 3 are realised by a regular dodecahedron. But finding a solution that satisfies all three conditions simultaneously turns out to be hard. A computer-assisted search only found a single example, up to Möbius transformations of the unit sphere:

The vertices are obtained from the following set of integer lattice points:

iverts = [[ 1840720, 1798335, 0],
[ 2573375, 0, 0],
[ 773500, 1963500, -1472625],
[ 773500, 1963500, 1472625],
[ 794070, 323510, -2426325],
[ 794070, 323510, 2426325],
[ 905828, -2223396, -926415],
[ 905828, -2223396, 926415],
[ 1925420, -863940, -1472625],
[ 1925420, -863940, 1472625],
[-1840720, -1798335, 0],
[-2573375, 0, 0],
[ -773500, -1963500, 1472625],
[ -773500, -1963500, -1472625],
[ -794070, -323510, 2426325],
[ -794070, -323510, -2426325],
[ -905828, 2223396, 926415],
[ -905828, 2223396, -926415],
[-1925420, 863940, 1472625],
[-1925420, 863940, -1472625]]

by dividing everything by 2573375. The planarity of the pentagonal faces and the fact that the points lie on a radius-2573375 sphere are checked by this notebook.

The dodecahedron has an order-8 symmetry group (it has three orthogonal planes of reflectional symmetry) and is visually very close to being regular. The symmetry group was a consequence of the search methodology; I specifically looked for solutions with this symmetry group in order to narrow down the search space.

By polar reciprocation, this also answers the equivalent question: ‘does there exist an icosahedron with rational vertices that contains an inscribed unit sphere?’

Methodology

Subject to the points being on the unit sphere, the planarity is equivalent to concyclicity. After stereographically projecting the unit sphere to the plane, which preserves rationality of points, the problem becomes:

Does there exist an embedding of the dodecahedral graph into the plane, with each vertex being a rational point, such that the five vertices of each face are all concyclic?

I restricted attention to symmetrical solutions invariant under reflection in each of the coordinate axes and under inversion in a circle centred on the origin:

(Note that the two points furthest from the origin are not shown in the above diagram.)

Here the red lines and circle are the axes of reflection and inversion, respectively. The three green circles are the necessary concyclicities, implying that the other nine pentagons are also concyclic by symmetry. These can be translated into Diophantine equations by using the fact that the four points {(x1, y1), (x2, y2), (x3, y3), (x4, y4)} are concyclic or collinear if and only if the following 4-by-4 determinant vanishes:

• 1, x1, y1, x1² + y1²;
• 1, x2, y2, x2² + y2²;
• 1, x3, y3, x3² + y3²;
• 1, x4, y4, x4² + y4²;

I wrote a program to search over integer values of a,b,c,d and check whether the values of x and y implied by the upper-right and upper-left green circles, respectively, are rational. If so, the program proceeds to check that the remaining (lower) green circle holds; if so, it reports a solution. The program reported several solutions, but they were all scalar multiples of a single ‘primitive’ solution and therefore correspond to the same polyhedron.

Do there exist any more solutions?

My search program (available here) only found a single primitive solution, even when the threshold was increased to N = 1000. This is not particularly conclusive: it’s often the case that Diophantine equations have solutions that are very few and far between, even if there are infinitely many solutions; the Pell equation is a simple example of this phenomenon.

EDIT: after running on a GPU with N = 10000, another two primitive solutions emerged.

Of course, if history is any guide, it’s entirely likely that Noam Elkies will storm in at any moment wearing a superhero cape and brandishing an infinite family of solutions to this problem.

In nested lattices, we talked about the E8 lattice and its order-696729600 group of origin-preserving symmetries. In minimalistic quantum computation, we saw that this group of 8-by-8 real orthogonal matrices is generated by a set of matrices which are easily describable in terms of quantum gates.

However, something that’s important to note is that quantum states are equivalence classes of vectors, where scalar multiples of vectors are identified. That is to say, the state space of n qubits is the complex projective space . So, if is a unitary matrix, then all unitary matrices of the form induce the same transformation.

Consequently, the order-696729600 subgroup of O(8) is not exactly the group of transformations in which we’re interested. Rather, we’re interested in its quotient by its centre {±I}. The resulting order-348364800 group G turns out to be very easy to describe without having to mention the E8 lattice!

G is isomorphic to the group of 8-by-8 matrices over the field of two elements which preserve the quadratic form

That is to say, each of the 256 binary vectors of length 8 can be classified as either:

• odd norm, if the number of ‘1’s in the vector is either 1 or 2 (modulo 4);
• even norm, if the number of ‘1’s in the vector is either 0 or 3 (modulo 4).

(Equivalently, the norm is the parity of c(c+1)/2, where c is the popcount (number of ‘1’s) in the binary vector.)

Then G is the group of invertible 8-by-8 binary matrices U which are norm-preserving, i.e. Ux has the same norm as x for all 256 of the choices of vector x.

Why?

To understand this isomorphism, we need to return to the description in terms of the E8 lattice, Λ. We’ll form a quotient of this lattice by a copy of the lattice scaled by a factor of two — the resulting set Λ / 2Λ contains 256 points which form an 8-dimensional vector space over the field of two elements!

Moreover, this set Λ / 2Λ consists of:

• 1 zero point (of even norm);
• 120 = 240 / 2 nonzero points of odd norm (each of which corresponds to a pair of antipodal vectors in the first shell of the E8 lattice);
• 135 = 2160 / 16 nonzero points of even norm (each of which corresponds to an orthoplex of 16 vectors in the second shell of the E8 lattice).

There isn’t an orthonormal basis for this set, so we choose the next best thing: a basis of 8 odd-norm vectors which are pairwise at 60º angles from each other! These, together with the origin, form the 9 vertices of a regular simplex in the E8 lattice. This choice of basis results in the norm having the simple expression we described earlier.

For concreteness, we’ll give an explicit set of vectors:

• e_0 = (½, ½, ½, ½, ½, ½, ½, ½);
• e_1 = (1, 1, 0, 0, 0, 0, 0, 0);
• e_2 = (1, 0, 1, 0, 0, 0, 0, 0);
• e_3 = (1, 0, 0, 1, 0, 0, 0, 0);
• e_4 = (1, 0, 0, 0, 1, 0, 0, 0);
• e_5 = (1, 0, 0, 0, 0, 1, 0, 0);
• e_6 = (1, 0, 0, 0, 0, 0, 1, 0);
• e_7 = (1, 0, 0, 0, 0, 0, 0, 1);

These satisfy 〈e_i, e_j〉 = 1 + δ_ij, where δ_ij is the Kronecker delta.

Computational convenience

In the article on minimalistic quantum computation, we mentioned that the choice of gates with dyadic rational coefficients were particularly convenient, as the matrix entries are exactly representable as IEEE 754 floating-point numbers. For the three-qubit case, this binary matrix representation is vastly more convenient still!

Firstly, an 8-by-8 binary matrix can be stored in a 64-bit processor register. Composing rotations is then reduced to multiplying these binary matrices.

Certain processor architectures support the operation of multiplying two 8-by-8 binary matrices in a single processor instruction! Knuth’s MMIX architecture calls this instruction ‘MXOR’, and there are at least two commercial computer architectures which also support this instruction: the original Cray supercomputer had this for government cryptographic reasons, and (up to one of the operands being transposed in the process) there is a vectorised implementation in the GFNI extension for x86_64. There was also a proposal for an extension for RISC-V with this instruction.

For instruction sets which don’t support MXOR in hardware, you can achieve the same effect with a bunch of bitwise manipulations. Challenge: try to implement MXOR in as few operations as possible using only bitwise Boolean operations and logical shifts. If I counted correctly, I can manage it in 113 total operations (73 Boolean operations and 40 shifts), but I expect it’s possible to do better than this.

What about inverting one of these matrices in G? It turns out that the exponent of G (the lowest common multiple of the orders of the elements) is 2520, so we can invert an element by raising it to the power of 2519. Using a technique which generalises repeated squaring, this can be accomplished by a chain of 15 MXOR instructions.

The previous post ended with unanswered questions about describing the Conway group, Co₀, in terms of quantum gates with dyadic rational coefficients. It turned out to be easier than expected, although the construction is much more complicated than the counterpart in the previous post.

We’ll start with a construction for a much smaller simple group — the order-168 group PSL(2, 7) — which can be generated by three CNOT gates acting on the three lower qubits in the diagram below:

For the moment, ignore the two unused upper qubits; their purpose will become clear soon enough. The three lower qubits have eight computational basis states (000, 001, 010, 011, 100, 101, 110, and 111). Viewing each of these states as a three-dimensional vector over , the CNOT gates induce linear shearing maps (and together generate the full 168-element group of linear automorphisms).

Now we introduce another two CNOT gates:

These act only on the two upper qubits, and generate the symmetric group (freely permuting the states 01, 10, and 11). These two gates, together with the three gates in the previous diagram, therefore generate the direct product of order 1008.

It is helpful to partition the 32 computational basis states into four 8-element sets:

• the set W consisting of all basis states of the form 00xyz;
• the set A consisting of all basis states of the form 01xyz;
• the set B consisting of all basis states of the form 10xyz;
• the set C consisting of all basis states of the form 11xyz;

where x, y, z are elements of {0, 1}. Then the two gates on the upper qubits are responsible for bodily permuting the three sets {A, B, C}; the three gates on the lower qubits induce the same linear permutation in each of W, A, B, and C (viewed as 3-dimensional vector spaces over the field of two elements).

Note that the permutation group is transitive on the 8-element set W, and transitive on the 24-element set V (the complement of W, or equivalently the union of A, B, and C), but no elements of V are ever interchanged with elements of W. This will remain the case as we continue to add further gates.

For the time being, we’ll therefore suspend thinking about the smaller 8-element set W, and concentrate on the larger 24-element set V.

We now introduce a sixth CNOT gate, bridging the upper and lower qubits:

This now expands the size of the group by a factor of 64, resulting in a 64512-element group called the trio group. The vector spaces A, B, and C are effectively upgraded into affine spaces which can be semi-independently translated (the constraint is that the images of their ‘origins’ — 01000, 10000, and 11000 — always have a modulo-2 sum of zero).

The trio group, considered as a permutation group on the 24 basis states in V, is a maximal subgroup of the simple sporadic group M₂₄. That means that adding a single further gate to break out of the trio group will immediately upgrade us to the 244823040-element Mathieu group! Unfortunately, I wasn’t able to find a particularly simple choice of gate:

The effect of this complicated gate is to do the following:

• Within each of the sets W and A, apply a particular non-affine permutation which exchanges the eight elements by treating them as four ‘pairs’ and swapping each element with its partner:
  • 000 ⇔ 100;
  • 001 ⇔ 110;
  • 010 ⇔ 111;
  • 011 ⇔ 101;
• Swap four elements of B with four elements of C by means of the Fredkin gate on the far right.

In terms of its action on V, this is an element of M₂₄, but does not belong to the trio group. Interestingly, it belongs to many of the other important subgroups of M₂₄ — namely PSL(3, 4) (also called M₂₁), the ‘sextet group’, and the ‘octad group’. At this point, the group generated by these seven gates is now the direct product of the alternating group A₈ (acting on the set W) and the Mathieu group M₂₄ (acting on the set V).

The elements of the Mathieu group are permutations on the set V, which can be viewed as 24-by-24 permutation matrices. Permutation matrices are matrices with entries in {0, 1}, where there’s exactly one ‘1’ in each row and column. If we throw in a Z-gate acting on the uppermost qubit, we’ll expand this to a group 2¹²:M₂₄ of ‘signed permutation matrices’ instead, where some of the ‘1’s are replaced with ‘−1’s. (The sets of rows and columns containing negative entries must form codewords of the binary Golay code; this is why each permutation matrix has only 2¹² sign choices instead of 2²⁴.) This group is interesting inasmuch as it’s the group of permutations and bit-flips which preserve the binary Golay code. It’s also a maximal subgroup, called the monomial subgroup, of the Conway group Co₀.

Instead of using this Z-gate, we’ll bypass the monomial subgroup and jump straight to the Conway group by adding a single additional three-qubit gate:

By a slight abuse of notation, we’ll reuse the symbols W and V to refer to the vector spaces spanned by first 8 basis states and final 24 basis states, respectively. After introducing this final gate, the resulting group of 32-by-32 matrices is a direct product of:

• an order-348364800 group (the orientation-preserving index-2 subgroup of the Weyl group of E₈) acting on the 8-dimensional vector space W;
• an order-8315553613086720000 group (the Conway group, Co₀) acting on the 24-dimensional vector space V.

These are the groups of rotational symmetries of the two most remarkable* Euclidean lattices — the E₈ lattice and the Leech lattice, respectively. Indeed, if we take all linear combinations (with integer coefficients!) of the images (under the group we’ve constructed) of a particular computational basis state, then we recover either the E₈ lattice or the Leech lattice (depending on whether we used one of the basis states in W or in V).

* as well as being highly symmetric, they give the optimal packings of unit spheres in 8- and 24-dimensional space, as proved by Maryna Viazovska.

These two groups each have a centre of order 2 (consisting of the identity matrix and its negation), modulo which they’re simple groups:

• the quotient of the order-348364800 group by its centre is PSΩ(8, 2);
• the quotient of Co₀ by its centre is the sporadic simple group Co₁.

This process of quotienting by the centre is especially natural in this quantum gate formulation, as scalar multiples of a vector correspond to the same quantum state.

Further connections

If we take qubits and the gates mentioned in the previous post, we remarked that the group generated is the full rotation group . If instead we replace the Toffoli gate in our arsenal with a CNOT gate, we get exactly the symmetry groups of the Barnes-Wall lattices! The orders of the groups are enumerated in sequence A014115 of the OEIS.

Acknowledgements

Thanks go to Conway and Sloane for their magnificent and thoroughly illuminating book, Sphere Packings, Lattices, and Groups. Also, the website jspaint.app (which is undoubtedly the most useful thing to have ever been written in JavaScript) was helpful for creating the illustrations herein.

In the usual ‘circuit model’ of quantum computation, we have a fixed number of qubits, {q₁, q₂, …, q_n}, and allow quantum gates to act on these qubits. The diagram below shows a Toffoli gate on the left, and an equivalent circuit of simpler gates on the right:

These diagrams represent qubits as horizontal lines (so there are 3 qubits in the circuit above), and the operations are applied from left to right. The circuit has 6 controlled-NOT gates (each acting on an ordered pair of qubits) and 9 single-qubit gates (4 T-gates, 3 inverse T-gates, and 2 Hadamard gates).

Whereas the internal state of a classical computer with n bits of memory can be described by a length-n vector of binary values, a quantum computer with n qubits of memory requires a length-(2^n) vector of complex numbers. A k-qubit gate is a unitary linear map described by a (2^k)-by-(2^k) matrix of complex numbers.

Importantly, it’s the exponential increase in the dimension of this vector (from n to 2^n), and not the involvement of complex numbers, which makes quantum computers [believed to be] able to solve more problems [in polynomial time] than is possible with a mere classical computer. To see this is the case, note that a (2^k)-by-(2^k) matrix of complex numbers can be emulated by a (2^(k+1))-by-(2^(k+1)) matrix of real numbers. Specifically, replace each complex entry with a real 2-by-2 block:

Consequently, a k-qubit complex gate can be emulated with a (k+1)-qubit real gate.

Indeed, it’s possible to restrict to not only real entries, but even to dyadic rational entries. Specifically, the most common universal set of logic gates {T, H, CNOT} consists of matrices whose entries belong to the ring where ζ is a primitive eighth root of unity. A similar trick means we can work over the ring of dyadic rationals instead, at the cost of just two extra qubits:

This is helpful for simulation in software: all finite values representable as IEEE 754 floating point numbers are dyadic rationals, and a partial converse is true: all dyadic rationals with numerator and denominator less than some bound (2^53 for double-precision and 2^24 for single-precision) are representable as IEEE 754 floating point numbers.

A universal pair of 3-qubit dyadic rational gates

Consider the Toffoli gate (left) and a new gate that I’m going to call the ‘XHH’ gate (it’s simply a tensor product of a Pauli X-gate and two Hadamard gates, all acting on separate qubits):

In an n-qubit circuit, each of these gates yields n(n – 1)(n – 2)/2 different matrices depending on the choice of qubits on which it acts, so this set expands to n(n – 1)(n – 2) matrices in total. Then we have the following universality theorem:

When n >= 4, the group generated by these matrices is a dense subgroup of the complete rotation group SO(2^n).

This is the best that we could hope for: when tensored by (n – 3) copies of the 2-by-2 identity matrix, these gates yield orthogonal matrices of determinant 1. It means that any special orthogonal gate can be approximated arbitrarily closely (in a number of gates polylogarithmic in the required precision, by the Solovay-Kitaev theorem), which (together with the above discussion of emulating complex unitary gates with real orthogonal gates) yields universal quantum computation.

An eight-dimensional surprise

More interesting is what happens when n = 3: the gates do not form a dense subgroup of O(8) as we might expect from extrapolating this result downwards (and noting that the Toffoli matrix has determinant -1, so the matrices lie in O(8) instead of SO(8) when n = 3).

Rather, they form a finite group of order 696729600.

This number should be familiar from the last post, because it’s the order of the E8 Weyl group. Every entry of every matrix in this group is not only dyadic rational, but in fact an integer multiple of 1/4. Inter alia, this finite group contains the familiar single-qubit Pauli gates X and Z, as well as the two-qubit CNOT gate.

Orthogonal projection of the 8-dimensional E8 root system into its 2-dimensional Coxeter plane, drawn by J. G. Moxness. The symmetry group of this set of 240 points is the aforementioned E8 Weyl group of order 696729600.

Given a starting state where all qubits are off (so the vector is (1, 0, 0, 0, 0, 0, 0, 0)), by applying these two gates in various combinations it is possible to reach any of 2160 different vectors (specifically, rescaled copies of the 2160 vectors in the second shell of the E8 lattice). If we instead began with an equal superposition of the eight computational basis states, there would be 240 reachable vectors — the root lattice vectors illustrated in the picture above! Again, the numbers 240 and 2160 should be very familiar from the previous article.

(Vectors that are related by scalar multiplication are identified as the same quantum state, so antipodal pairs of vectors in the previous paragraph correspond to the same quantum state. Consequently, there are only 1080 distinct quantum states reachable from the all-off quantum state, or 120 distinct quantum states reachable from the )

Before I found the set {Toffoli, XHH}, I tried the deceptively similar pair {Toffoli, ZHH}. To my surprise, the program I used to compute the group generated by those elements had an order of 2903040 — significantly smaller than the 696729600-element group I had expected! This is the Weyl group of E7, and in this case we get the smaller subgroup because all six matrices fix the vector (1, 1, 1, -1, 1, -1, -1, -1) and therefore only act nontrivially on its seven-dimensional orthogonal complement. Fortunately, the set {Toffoli, XHH} does not have a fixed vector and generates the full Weyl group of E8.

Unanswered questions

The {Toffoli, ZHH} construction of the E7 Weyl group demonstrates that we can realise the origin-preserving isometry group of a 7-dimensional lattice, even though 7 is not a power of two. An even more beautiful and exceptional lattice than the E8 lattice is the 24-dimensional Leech lattice, whose origin-preserving symmetry group is the Conway group Co0. Is there an elegant set of matrices which generate a group isomorphic to Co0 and have a simple description in terms of quantum gates?

Edit: depending on whether you’d consider it elegant and/or simple, there’s an affirmative answer in the next post.

The first nonempty shell of the Leech lattice consists of 196560 points, as opposed to the 240 points in the first shell of the E8 lattice. David Madore has plotted some beautiful projections of this set — they’re as close as possible to being analogous to the Petrie projection of the E8 root system shown above, except inasmuch as Co0 is not a Coxeter group and therefore it’s unclear which plane (if any!) is analogous to the Coxeter plane.

1, 240, 2160, 6720, 17520, 30240, 60480, 82560, 140400, …

These terms count the number of points at distance from the origin in the E8 lattice, a highly symmetric arrangement of points which Maryna Viazovska recently (in 2016) proved is the densest way to pack spheres in 8-dimensional space.

Even more recently, Warren D. Smith noticed a rather exceptional numerical coincidence — the sum of the first three terms in this sequence is a perfect fourth power:

1 + 240 + 2160 = 2401 = 7^4

Is this merely a coincidence? To begin with, we’ll look at the E8 lattice through a different lens: as a subset not of , but of complex (alas, not projective) 4-space, . Specifically, recall the Eisenstein integers, the ring of complex numbers generated by a cube root of unity:

These points have been 3-coloured according to whether the Eisenstein integer is congruent (modulo ) to 0, +1, or −1. The E8 lattice is then concisely expressible as the set of points (w, x, y, z) where:

• each coordinate is an Eisenstein integer;
• the colours of the four coordinates form a codeword in the tetracode.

The tetracode is a 2-dimensional subspace of the 4-dimensional vector space . A point belongs to the tetracode if and only if w = y − x = z − y = x − z. Equivalently, the coordinates are of the form (d, a, a + d, a − d).

The real E8 lattice has a symmetry group of order 696729600; this complex E8 lattice has more structure (complex scalar multiplication) which reduces the symmetry group to order 155520. This extra structure is what we need to explain Warren’s coincidence.

Now that we have defined the E8 lattice as a complex 4-dimensional lattice, consider scalar multiplication by 3 + ω. Returning to an 8-dimensional real viewpoint, this linear map corresponds to composing a rotation by a scaling by sqrt(7), and therefore has a determinant of sqrt(7)^8 = 2401. The image of the E8 lattice under this map is a sublattice, geometrically similar to the original E8 lattice, and the Voronoi cells induced by this sublattice each consist of 2401 points from the original lattice (namely one central point, surrounded by an inner shell of 240 points and an outer shell of 2160 points).

Linear subspaces and positional number systems

If we sum the first three terms of the theta series of the E6 lattice, 1 + 72 + 270, we get exactly 7^3. And similarly for the D4 lattice, 1 + 24 + 24 = 7^2. For the A2 (hexagonal) lattice, we have 1 + 0 + 6 = 7^1. These correspond to taking 3-, 2-, and 1-dimensional (complex) linear subspaces of the Voronoi partition described above.

The latter is particularly easy to describe: every Eisenstein integer z can be uniquely expressed in the form (3 + ω)q + r, where q is an Eisenstein integer and r belongs to the set {0, 1, ω, ω², −1, −ω, −ω²} consisting of zero together with the sixth roots of unity. We can decompose q in the same manner, and continue recursively; this is tantamount to writing an Eisenstein integer in a positional number system with radix 3 + ω and the seven digits {0, 1, ω, ω², −1, −ω, −ω²}.

If we allow digits after the ‘decimal point’, we can express any complex number in this positional number system. The set of numbers which ’round to zero’ after discarding the digits after the decimal point form the Gosper island, a self-similar set with a fractal boundary:

There are two different principal cube roots of unity, and each one yields a different (albeit isomorphic) positional number system for complex numbers.

The Hurwitz integral quaternions form a scaled copy of the D4 lattice, and we similarly obtain a positional number system for the quaternions with radix 3 + ω and 49 different digits: the Hurwitz integral quaternions with squared norm at most 2. There are 8 principal cube-roots of unity, and each one determines a positional number system for quaternions. It’s worth commenting that this isn’t the only nice* number system for integral quaternions: radix-(2 + i) with 25 digits (the Hurwitz integers of norm at most 1) also works.

*where ‘nice’ is defined to mean that the digit set consists of all integers with norm <= r for some value of r.

Finally, the Cayley integer octonions form a scaled copy of the E8 lattice, and there are 56 principal cube-roots of unity. Any one of these results in a positional number system for the octonions (which is well-defined despite the non-associativity of the octonions, since any pair of octonions together generate an associative algebra) with radix 3 + ω and 2401 different digits: the integer octonions with squared norm at most 2. We’ll call this number system ‘Warrenary’ after its discoverer. Unlike in the real, complex, and quaternionic cases, Warrenary is the unique positional number system for the octonions satisfying the aforementioned niceness property.

There is no analogue in six dimensions: normed division algebras only exist in dimensions 1, 2, 4, and 8, so there is no positional number system corresponding to the recursive nesting of E6 lattices.

Disappointingly, there appears not to be any similarly elegant lattice nestings for higher-dimensional lattices beyond D4, E6 and E8, such as the 24-dimensional Leech lattice (in particular, no early initial segment of the theta series yields a perfect 12th power as its sum). As such, the recursive nesting of E8 lattices is quite exceptional indeed.

Further reading

The integral quaternions and octonions have many other fascinating and elegant properties, including analogues of unique prime factorisation, which are explained in the book On Quaternions and Octonions by Derek Smith and the late, great John Conway (1937 — 2020).

Positional number systems for the real and complex numbers are described in the Seminumerical Algorithms volume of Donald Knuth’s The Art of Computer Programming.

In a previous article, an announcement was made of a complex self-replicating machine (known as the 0E0P metacell) in a simple 2-state cellular automaton. In the interim between then and now, Thomas Cabaret has prepared a most illuminating video* explaining the method with which the machine copies itself:

[youtube https://www.youtube.com/watch?v=CfRSVPhzN5M]

Note: the video is in French; recently, Dave Greene added an English translation of the subtitles.

* the video is part of Cabaret’s Passe-Science series. You may enjoy some of his other videos, including an explanation of the P vs NP problem and a reduction of Boolean satisfiability to the 3-colourability of planar graphs.

Anachronistic self-propagation

In related news, Michael Simkin recently created a wonderfully anachronistic self-propagator entitled Remini: it uses the same single-channel/slow-salvo construction mechanism as the 0E0P metacell, but it is built from oscillatory components instead of static ones. That is to say, it implements modern ideas using components available in the 1970s.

The project involved slmake together with a suite of additional tools developed by Simkin. There isn’t a video of this machine self-replicating, so you’d need to download a program such as Golly in order to watch it running.

Further reading

For further reading, I recommend (in order):

• The wiki entry (under construction) for the 0E0P metacell;
• An article unveiling various simpler examples of self-constructing circuitry;
• The slmake repository;
• A tutorial on effective use of slmake;
• A challenge thread proposing another contraption, that no-one has yet built. This would require the use of slmake followed by some ‘DNA-splicing’ to interleave the construction recipe with extra operations.