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How do four groups build E₈?

Build the pattern from groups of 52, 60, 64 and 64 points. Each point represents one of E₈’s shortest nonzero vectors, called roots.

Step 1 of 8

Start with the two specified root lattices

The carrier side supplies D₅ and the family side supplies A₃. Their ranks are five and three. Keep both integral pairings: joining their eight directions requires more than adding the counts.

1. The positive definite D₅ and A₃ root lattices with their standard integral forms, normalized so roots have squared length two.. Form their orthogonal direct sum.. L₀ = D₅ ⊕ A₃ has rank eight and determinant 4 × 4 = 16.. 2. The two lattice forms and their chosen generating classes in the dual lattices modulo the original lattices.. Evaluate the discriminant quadratic forms and add the generating norms.. q_D(1) + q_A(1) = 5/4 + 3/4 = 2.. 3. The two cyclic discriminant groups and the compatible generating norms.. Match k on one side with k on the other, for k = 0, 1, 2, 3.. H = ⟨(1,1)⟩ has order four and is isotropic for the discriminant form.. 4. The rank eight direct sum with determinant sixteen and the isotropic subgroup H of order four.. Take the associated even overlattice and apply the determinant versus index formula.. The new lattice is even, positive definite, rank eight and unimodular.. 5. The even extension and its standard five plus three Euclidean coordinate realization, with A₃ represented isometrically by D₃.. Enumerate the signed pairs within blocks and across the block boundary.. The integer groups contain 52 and 60 roots, totalling 112.. 6. The same coordinate extension and its two matched spinor cosets.. Enumerate five-sign and three-sign blocks with matching parity.. The half integer groups contain 64 and 64 roots, totalling 128.. 7. The two enumerated integer groups and two enumerated half integer groups.. Join the lists, remove no vectors, and evaluate each vector’s Euclidean squared norm.. 240 distinct roots, each with squared length two.. 8. The positive rank eight extension, determinant one and the constructed list of roots.. Form the simple root Gram matrix and check the original lattice certificate.. The E₈ lattice: rank 8, even, determinant 1; its Lie algebra has 240 + 8 = 248 dimensions..
Arrows carry the named inputs into later calculations. Added inputs stay marked. Reduced motion is on. Use Next step or choose a stage below.
Carry forward
The positive definite D₅ and A₃ root lattices with their standard integral forms, normalized so roots have squared length two.
Added at this step
Begin with the specified positive root lattices and their actual integral pairings.
Operation
Form their orthogonal direct sum.
Result
L₀ = D₅ ⊕ A₃ has rank eight and determinant 4 × 4 = 16.
L0=D5⊕A3,rank⁡L0=5+3=8,det⁡L0=4⋅4=16\begin{gathered}L_0=D_5\oplus A_3,\\[.4em]\operatorname{rank}L_0=5+3=8,\\[.4em]\det L_0=4\cdot4=16\end{gathered}

In words: Form their orthogonal direct sum.

Why this step works and what it assumes

D₅ is the stated carrier root lattice. A₃ is the family residue lattice, with its integral Cartan pairing. Its pairing is distinct from the analytic Green metric between marked points. These are internal lattice directions, rather than a count of physical spacetime dimensions.

Read the original calculation
See where this fits in the whole theory
Try the individual controls

Try this. Press Build next group until all 240 roots appear. Use Replay to watch the four additions again.

Join four groups of points

Build one group at a time. Each dot represents one shortest vector in the eight-coordinate lattice. The four groups together contain 240 vectors.

52Group 160Group 264Group 364Group 4
The movement assembles a drawing. The dots are mathematical vectors, rather than moving physical particles. Different vectors can overlap on this flat screen.
0 of 240 points added? + ? + ? + ?

This is a flat projection of the original E₈ vectors. The full calculation explains why these four particular groups fit into one even lattice with determinant one.

Begin with the first group

Add the first fifty two roots. They come from the two starting patterns, with forty on the carrier side and twelve on the family side.

Add the missing groups

Add sixty mixed roots, then two groups of sixty four. The running total reaches 240.

Look at the same pattern again

Replay the additions. Each group shows another part of the same fixed E₈ pattern.

Explore the full interactive calculation

Inspect all 240 roots

A lattice is a regular pattern of points. A root is one of its shortest nonzero vectors, measured from the origin to a lattice point. Each dot below shows one of E₈’s 240 root vectors, projected from eight coordinates into two.

The four glue classes are the four groups of vectors joined by this construction. Choose a group, then select a dot or move to the next root. The white ring follows the selected vector; its connecting lines show a precise lattice relationship.

All 240 vectors have squared length two. The four cosets contain 52, 60, 64 and 64 roots. Use the controls below to inspect their coordinates.
The projection turns; the lattice stays fixed. Overlapping points are separate roots. Eight lattice directions describe internal geometry, rather than eight spacetime dimensions.
Rotation paused by your motion preference.

Selected root · class 0

D₅ · 1-1D₅ · 2-1D₅ · 30D₅ · 40D₅ · 50A₃ · 10A₃ · 20A₃ · 30

Squared length 2 · 56 neighbours · 52 + 60 + 64 + 64 = 240 roots

What exactly has been drawn?

The integer roots are ±eᵢ ±eⱼ for i < j. The half integer roots have eight entries ±½ and an even number of minus signs. Two orthonormal projection directions rotate through a four dimensional subspace of the eight coordinate space. Class selection highlights the original D₅ ⊕ D₃ coordinate split, with A₃ isometric to D₃. A line joins the selected root to one of its 56 neighbours whose inner product with it is one. A projection preserves neither every visible distance nor every separation.

R(E8)={±ei±ej:i<j}∪{12(ϵ1,…,ϵ8):∏iϵi=1}R(E_8)=\{\pm e_i\pm e_j:i<j\}\cup\{\tfrac12(\epsilon_1,\ldots,\epsilon_8):\prod_i\epsilon_i=1\}
Inspect the original lattice certificate

How this fits into the whole theory

The carrier and family supply two starting patterns. The four groups show how their specified join builds the E₈ pattern.

What this experiment establishes

This is a lattice theorem for the stated D₅ and A₃ inputs and their compatible glue. Identifying its representations and operators with a physical field theory requires the corresponding field and source dictionary.