Since ancient times people have written, drawn and painted using pigments and dyes in water, often without any binder to adhere them to the ground. These are generically termed inks, and distinct from other media such as watercolours or the temperas. Most common among them is India ink, whose essential ingredients are carbon particles from soot, suspended in water. When applied to a suitably absorbent ground that has stood the test of centuries.

Hieronymus Bosch’s study of The Owl’s Nest from around 1505-1516 is a good example of a pen and ink drawing starting to transcend into painting.

Washes applied using a brush became common in sketches made in front of the motif, such as Egbert van der Poel’s Fire in the Nieuwe Kerk, Amsterdam, in 1645. Known for his paintings of fires, he used these sketches to paint his famous brandjes in the studio. Landscape artists such as Nicolas Poussin and Claude Lorrain used the same sketching media working en plein air to build image libraries for the idealised landscapes used in their finished paintings.

By the early nineteenth century, artists like Samuel Palmer were creating works that are to all intents and purposes full-blown paintings using black and coloured inks, such as this atmospheric nocturne of Cornfield and Church by Moonlight from about 1830.

Used without a binder, inks normally rewet and smudge or blur, so have to be applied after any watercolour. Adding a binder such as shellac to India ink makes it waterproof when it has dried, and this was used in combination with watercolours by illustrators and painters such as the great Arthur Rackham, here in one of his illustrations for an edition of Charles Dickens’ A Christmas Carol.

A related medium that also crossed over from drawings and illustrations into paintings is casein, a protein-based tempera. It too has a long history, but didn’t become popular among artists until the late nineteenth century. Casein is a protein originally obtained from sour milk, but by the nineteenth century it was more usually prepared by the addition of rennet, enzymes used in cheese manufacture, extracted from the stomachs of calves.

Casein powder is then turned into paint by dispersion in an alkaline solution, typically made from lime or borax, and pigment is ground in. When lime is used the paint works best on porous grounds, but must be used fresh. When a little linseed oil is added to borax casein, a shelf-life of several months or more is possible. Casein paints dry as quickly as egg tempera, but can be reworked for a period until their binder has fully hardened. Once that has taken place, their paint film is robust and stable.

The earliest painting I have seen that is known to have been made using casein paints is William Morris Hunt’s unusual view of Niagara from 1879. Hunt deviated here from his customary use of oils, although he still applied his paint to a prepared canvas support.

One of the most famous artists to have used casein paint is Gustav Klimt. In 1902, he used it in a frieze of 24 metres (nearly 80 feet) length for the fourteenth exhibition of the Vienna Secession, his Beethoven Frieze, of which that above is a section known as The Hostile Powers, and that below is Nagging Grief. He applied his casein paint directly onto mortar with added stucco, gold leaf, and other materials. This is known to be a robust form of wall-painting that is much more accommodating than fresco, and just as enduring, but has never become popular.

A few of Colin Campbell Cooper’s skyscraper cityscapes were painted using casein media, including this view of Flatiron Building, Manhattan from about 1908. He was equally proficient in watercolour and oils, and I don’t know why he experimented with casein, nor why he used it so infrequently.

Another Vienna-based artist, Albin Egger-Lienz, also used casein on occasion, here in this fourth of several different versions of Totentanz (Dance of Death) from 1915.

During the twentieth century, casein paints didn’t become popular but continued in use, particularly by those who also made illustrations. Just when they were attracting a following among artists such as Andy Warhol, they were rapidly displaced by acrylics. Despite that, casein paint is still available today, sold by Pelikan under the brand name of Plaka.

Every self-respecting file system identifies files and directories using numbered data structures. In most modern file systems, those data structures are known as inodes, and their numbers are inode numbers, sometimes shortened to inodes. The term is thought to be a contraction of index node, which certainly makes sense, but is lost in the mists of time.

In any file system, for example an individual APFS volume, the inode numbers uniquely identify each inode, and each object within that file system has its own inode. Whatever else the file system might do, the inode number identifies one and only one object within it. Thus one invariant way of identifying any file is by referencing the file system containing it, and its inode number.

HFS+

The Mac’s original native file systems, ending most recently in Mac OS Extended File System, or HFS+ from its origins in the Hierarchical File System, don’t use inodes as such, and don’t strictly speaking have inode numbers. Instead, the data structures for their files and folders are kept in Catalogue Nodes, and their numbers are Catalogue Node IDs, CNIDs.

With Mac OS X came Unix APIs, and their requirement to use inodes and inode numbers. As CNIDs are unique to each file and folder within an HFS+ volume, for HFS+ they are used as inode numbers, although in some ways they differ. CNIDs are unsigned 32-bit integers, with the numbers 0-15 reserved for system use. For example, CNID 7 is normally used as the Startup file’s ID.

Although not necessarily related to CNIDs, it’s worth noting that the Mac’s original file system MFS allowed a maximum of 4,094 files, its successor HFS was limited to 65,535, and HFS+ to 4,294,967,295. Those are in effect the maximum number of inode numbers or their equivalents allowed in that file system.

APFS

Unlike HFS+, APFS was designed from the outset to support standard Posix features, so has inodes numbered using unsigned 64-bit integers.

Strictly, the APFS inode number is the object identifier in the header of the file-system key. Not all unsigned 64-bit integers can be used as inode numbers, though, as a bit mask is used, and the number includes an encoded object type. APFS also makes special allowance for volume groups consisting of firmlinked System and Data volumes, to allow for their inode numbers to remain unique across both volumes. Officially, those allow for a maximum number of inode numbers of 9,223,372,036,854,775,808, either in single APFS file systems, or shared across two in a volume group.

Every file in APFS is required to have, at an absolute minimum, an inode and its associated attributes, shown above in blue, including a set of timestamps, permissions, and other essential information about that file.

There are also three optional types of component:

• although some files may be dataless, most have data, for which they need file extents (green), records linking to the storage blocks containing that file’s data (pink);
• smaller extended attributes (yellow), named metadata objects that are stored in a single record;
• larger extended attributes (yellow), over 3,804 bytes in size, whose data is stored separately in a data stream.

Interpretation

When looking at inode numbers in volume groups, they can be used to determine which of the firmlinked file systems contains any given file. Both volumes share the same volume ID, and files in the System volume have very high inode numbers, while those in the Data volume are relatively low. I have given further details here.

Inode numbers are more generally useful in distinguishing files whose data appears identical, and the behaviour of various methods of linking files.

Copying a file within the same file system (volume) creates a new file with its own inode number, of course. However, duplicating a file within the same file system results in an APFS clone file, with a distinct inode, although it shares common data, so their inode numbers are also different.

Instead of duplicating everything, only the inode and its attributes (blue and pink) are duplicated, together with their file extent information. There’s a flag in each clone file’s attributes to indicate that cloning has taken place.

A symbolic link or symlink is merely a pointer to the linked file’s path, and doesn’t involve inode numbers at all. Because of that, changing the linked file’s name or path breaks its link. Finder aliases and their kindred bookmarks do contain inode numbers, so should be able to cope with changes to the linked file’s name or path, provided it remains in the same file system and doesn’t change inode number.

Hard links are more complex, and depend on the way in which the file system implements them, although the fundamental rule is that each object that’s hardlinked to the same file has the same inode number. According to Apple’s reference to APFS, this is how it handles hard links.

When you create a hard link to a file (blue), APFS creates two siblings (purple) with their own IDs and links, including different paths and names as appropriate. Those don’t replace the original inode, and there remains a single file object for the whole of that hardlinked file.

Inode attributes keep a count of the number of links they have to siblings in their link (or reference) count. Normally, when a file has no hard links that’s one, and there are no sibling files. When a file is to be deleted, if its link count is only 1, the file and all its associated components can be removed, subject to the requirements of any clones and applicable snapshots. If the link count is greater than 1, then only the sibling being removed is deleted.

Easy access

Inode numbers are readily accessed using command tools in Terminal. For example, ls -i lists items with their inode numbers shown. One free utility that displays full information about inode numbers and much more is Precize.

The volume ID is given as the first number in the volfs path, and the second is the inode number of that file within that. Note that the File Reference URL (FileRefURL) uses a different numbering system, and the Ref count of 1 indicates this file has no hard links.