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Memory geometry

Memory geometry is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Memory geometry rather than just read about it. In short: In the design of modern computers, memory geometry describes the internal structure of random-access memory. Memory geometry is of concern to consumers upgrading their computers, since older memory controllers may not be compatible with later products.

Memory geometry — main illustration
Memory geometry — illustration

Key takeaways

  • Memory geometry belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Memory geometry to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Memory geometry from memory before moving on to harder problems.

Reference excerpt

In the design of modern computers, memory geometry describes the internal structure of random-access memory. Memory geometry is of concern to consumers upgrading their computers, since older memory controllers may not be compatible with later products. Memory geometry terminology can be confusing because of the number of overlapping terms. The geometry of a memory system can be thought of as a multi-dimensional array. Each dimension has its own characteristics and physical realization. For example, the number of data pins on a memory module is one dimension.

Physical features

Memory geometry describes the logical configuration of a RAM module, but consumers will always find it easiest to grasp the physical configuration. Much of the confusion surrounding memory geometry occurs when the physical configuration obfuscates the logical configuration. The first defining feature of RAM is form factor. RAM modules can be in compact SO-DIMM form for space constrained applications like laptops, printers, embedded computers, and small form factor computers, and in DIMM format, which is used in most desktops. The other physical characteristics, determined by physical examination, are the number of memory chips, and whether both sides of the memory "stick" are populated. Modules with the number of RAM chips equal to some power of two do not support memory error detection or correction. If there are extra RAM chips (between powers of two), these are used for ECC. RAM modules are 'keyed' by indentations on the sides, and along the bottom of the module. This designates the technology, and classification of the modules, for instance whether it is DDR2, or DDR3, and whether it is suitable for desktops, or for servers. Keying was designed to make it difficult to install incorrect modules in a system (but there are more requirements than are embodied in keys). It is important to make sure that the keying of the module matches the key of the slot it is intended to occupy. Additional, non-memory chips on the module may be an indication that it was designed for high capacity memory systems for servers, and that the module may be incompatible with mass-market systems. As the next section of this article will cover the logical architecture, which covers the logical structure spanning every populated slot in a system, the physical features of the slots themselves become important. By consulting the documentation of your motherboard, or reading the labels on the board itself, you can determine the underlying logical structure of the slots. When there is more than one slot, they are numbered, and when there is more than one channel, the different slots are separated in that way as well – usually color-coded.

Logical features In the 1990s, computers using cache-coherent non-uniform memory access were released, which allowed combining multiple computers that each had their own memory controller such that the software running on them could use I/O devices, memory, and CPU of all participating systems as if they were one unit (single system image). With AMD's release of the Opteron, which integrated the memory controller into the CPU, NUMA systems that share more than one memory controller in a single system have become common in applications that require the power of more than the common desktop. Channels are the highest-level structure at the local memory controller level. Modern computers can have two, three or even more channels. It is usually important that, for each module in any one channel, there is a logically identical module in the same location on each of the other populated channels. Module capacity is the aggregate space in a module measured in bytes, or – more generally – in words. Module capacity is equal to the product of the number of ranks and the rank density, and where the rank density is the product of rank depth and rank width. The standard format for expressing this specification is (rank depth) Mbit × (rank width) × (number of ranks). Ranks are sub-units of a memory module that share the same address and data buses and are selected by chip select (CS) in low-level addressing. For example, a memory module with 8 chips on each side, with each chip having an 8-bit-wide data bus, would have one rank for each side for a total of 2 ranks, if we define a rank to be 64 bits wide. A module composed of Micron Technology MT47H128M16 chips with the organization 128 Mib × 16, meaning 128 Mi memory depth and 16-bit-wide data bus per chip; if the module has 8 of these chips on each side of the board, there would be a total of 16 chips × 16-bit-wide data = 256 total bits width of data. For a 64-bit-wide memory data interface, this equates to having 4 ranks, where each rank can be selected by a 2-bit chip select signal. Memory controllers such as the Intel 945 Chipset list the configurations they support: "256-Mib, 512-Mib, and 1-Gib DDR2 technologies for ×8 and ×16 devices", "four ranks for all DDR2 devices up to 512-Mibit density", "eight ranks for 1-Gibit DDR2 devices". As an example, take an i945 memory controller with four Kingston KHX6400D2/1G memory modules, where each module has a capacity of 1 GiB. Kingston describes each module as composed of 16 "64M×8-bit" chips with each chip having an 8-bit-wide data bus. 16 × 8 equals 128, therefore, each module has two ranks of 64 bits each. So, from the MCH point of view there are four 1 GB modules. At a higher logical level, the MCH also sees two channels, each with four ranks. In contrast, banks, while similar from a logical perspective to ranks, are implemented quite differently in physical hardware. Banks are sub-units inside a single memory chip, while ranks are sub-units composed of a subset of the chips on a module. Similar to chip select, banks are selected by bank select bits, which are part of the memory interface.

Hierarchy of organization

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Memory geometry

Start with the simplest possible case. Write down what Memory geometry claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Memory geometry before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Memory geometry ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Memory geometry

In research
Memory geometry appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Memory geometry in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Memory geometry is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computer memory, Computing terminology, so understanding it makes those chapters shorter.
In everyday life
Look for Memory geometry outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Memory geometry in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Memory geometry means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Memory geometry out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Memory geometry in simple terms?

In the design of modern computers, memory geometry describes the internal structure of random-access memory. Memory geometry is of concern to consumers upgrading their computers, since older memory controllers may not be compatible with later products.

Why does Memory geometry matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Memory geometry?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Memory geometry.

Tags

  • Computer memory
  • Computing terminology

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