A lattice is a discrete repeat of space
By now you are fluent with a convex body K in R^n — a compact convex set with nonempty interior — and with the Brunn-Minkowski inequality that controls how its volume behaves under Minkowski sums. This guide pivots to a second, completely rigid object living in the same space: a lattice. Fix a basis b_1, ..., b_n of R^n; the lattice L is the set of all integer combinations m_1 b_1 + ... + m_n b_n with each m_i in Z. The picture to hold is the standard grid Z^n in R^2, but sheared: not the squares of graph paper necessarily, but any regularly repeating array of dots with no accumulation point.
The single most important number attached to L is its covolume (or determinant) det(L) = |det(B)|, where B is the matrix whose columns are the b_i. Geometrically det(L) is the volume of the fundamental parallelepiped {sum t_i b_i : 0 <= t_i < 1}: one tile of the repeating pattern. Translates of that tile by lattice vectors fill R^n exactly once, with no overlaps and no gaps — so det(L) is the volume of space per lattice point. Different bases of the same L give different-looking tiles but always the same covolume, because changing basis multiplies B by an integer matrix of determinant +/- 1.
Minkowski's theorem: convexity plus symmetry plus enough volume
Here is the centerpiece. Minkowski's lattice point theorem says: let L be a lattice in R^n and let K be a convex body that is symmetric about the origin (so x in K implies -x in K). If vol(K) > 2^n det(L), then K contains a lattice point other than 0. The hypotheses are not decoration — drop any one and the conclusion fails. The statement is the heart of Minkowski's theorem, and the whole subject of the geometry of numbers grows out of squeezing it.
Why exactly 2^n? Look at the half-size body (1/2)K. Its volume is vol(K)/2^n, which the hypothesis makes strictly bigger than det(L). Now picture each translate of the fundamental tile and fold (1/2)K back into one tile mod L. If (1/2)K had no two points differing by a lattice vector, the folding would be injective and (1/2)K would fit inside one tile of volume det(L) — impossible, since its volume is larger. This is a pigeonhole argument by volume, sometimes called Blichfeldt's lemma. So there exist distinct x, y in (1/2)K with x - y in L.
The finish uses the two hypotheses we have not spent yet. Since x, y are in (1/2)K, the points 2x and 2y lie in K. By central symmetry -2y is in K, and by convexity the midpoint (2x + (-2y))/2 = x - y is in K. But x - y is a nonzero lattice vector. Done. Notice precisely where each hypothesis entered: symmetry gave us -2y, convexity gave us the midpoint, and the volume bound 2^n det(L) drove the pigeonhole. Strip out symmetry and a thin nonsymmetric sliver can dodge every lattice point; strip out convexity and a measurable but bent region can too.
Two classic harvests: sums of squares and Dirichlet
The magic is that a purely geometric existence theorem produces arithmetic. Take Fermat's two-squares theorem: every prime p with p = 1 mod 4 is a sum of two squares. Number-theoretic input gives an integer u with u^2 = -1 mod p. Form the lattice L spanned by (1, u) and (0, p); its covolume is the determinant of the matrix [1, 0; u, p], namely p. Every lattice vector (a, b) satisfies a^2 + b^2 = 0 mod p, because b = a u + p k forces a^2 + b^2 = a^2(1 + u^2) mod p = 0. Now apply Minkowski.
Choose K to be the open disk of radius r centered at 0, a convex and centrally symmetric body of area pi r^2. We want a nonzero lattice point inside, so we need pi r^2 > 2^2 det(L) = 4p, i.e. r^2 > 4p/pi. Pick r^2 just above 4p/pi but still below 2p (legal since 4/pi < 2). Minkowski hands us a nonzero (a, b) in L with a^2 + b^2 < 2p. But a^2 + b^2 is a positive multiple of p, and the only positive multiple of p strictly below 2p is p itself. Hence a^2 + b^2 = p. The geometry of a disk just factored a prime.
The same engine drives Dirichlet's theorem on simultaneous rational approximation, where the symmetric convex body is a thin tall box (a slab cross product) rather than a disk. The recipe is always the same three moves, and learning to run it backwards — from the arithmetic you want, to the lattice and the body that deliver it — is the real skill of the geometry of numbers.
Three moves, every time:
1. choose lattice L -> det(L) encodes the arithmetic constraint
2. choose body K -> symmetric + convex, vol(K) > 2^n det(L)
3. Minkowski => exists 0 != v in K cap L
two-squares: L = <(1,u),(0,p)>, det = p, K = disk pi r^2 > 4p
=> a^2 + b^2 = pSuccessive minima and the shape of a lattice
Minkowski's first theorem is about one short vector. His second theorem measures the full shape. Fix a symmetric convex body K and define the i-th successive minimum lambda_i(K, L) as the smallest scale t > 0 such that tK contains i linearly independent lattice vectors. So lambda_1 is the radius at which K first touches a nonzero lattice point — the shortest-vector scale — and lambda_1 <= lambda_2 <= ... <= lambda_n grows as you demand more independent directions. These numbers refuse to be read off any single basis; they are intrinsic invariants of how L sits inside the metric of K.
Minkowski's second theorem then sandwiches their product: (2^n / n!) det(L) <= lambda_1 lambda_2 ... lambda_n vol(K) <= 2^n det(L). The upper bound is the deep half; the lower bound is an easy packing count. Read it as a conservation law: a lattice cannot be short in every direction at once. If lambda_1 is tiny — there is a very short vector — then some later lambda_i must be correspondingly large, because their product is pinned near det(L)/vol(K) up to the 2^n / n! window. This is the precise sense in which the first guide's intuition, that convexity trades volume against extent, becomes a statement about integers.
From lattices to packing and Voronoi cells
Center a ball of radius lambda_1/2 at every lattice point. Because lambda_1 is the shortest distance between distinct lattice points, these balls have disjoint interiors: you have built a lattice sphere packing. Its density is (volume of one ball) / det(L), since each fundamental tile carries exactly one center. This is the cleanest bridge from the geometry of numbers to sphere packing: a good lattice — large lambda_1 relative to det(L) — is a dense packing, and the search for record packings in dimensions 8 and 24 is exactly the search for extraordinary lattices (E_8 and the Leech lattice).
Dual to packing is tiling. Assign each point of R^n to its nearest lattice point; the region claimed by 0 is the Voronoi cell, a convex centrally symmetric polytope, and its translates tile space — a better-fitting tile than the parallelepiped. You met the Voronoi diagram in the discrete-geometry rung; here it is the natural fundamental domain, so vol(Voronoi cell) = det(L) too. The covering radius (largest distance from any point of R^n to L) and the packing radius lambda_1/2 are tied to the Voronoi cell's outradius and inradius, and their ratio quantifies how round, hence how efficient, the lattice is.
One honest caveat to close. Minkowski's theorems and successive minima are about lattice packings; the genuinely hardest packing questions allow centers anywhere, not on a lattice. In most dimensions the densest packing need not be a lattice at all, and we simply do not know it — the optimal density is settled only in dimensions 1, 2, 3, 8, and 24. So treat the lattice story as a powerful, fully proved special case, and resist the temptation to call it the whole of sphere packing. As elsewhere on this ladder, the proved special case is worth more than a slogan about the general one.