Microscopists in the 19th century were fascinated by frustules , the patterned siliceous coatings of diatoms, varying from one species to another (see below). When the algae die, frustules do not readily decompose and they fall to the sea, or lake, bed, joining those from diatoms that live there.
Experts arranged the frustules of different species on slides to produce kaleidoscopic patterns and this art form continues today in the wonderful work of Klaus Kemp  and others. However, it wasn't just the frustules that interested the many microscopists among the developing middle classes, as living diatoms were also observed, including those that move by gliding. This movement was described by Philip Henry Gosse  in one of his books that popularised Natural History and microscopy;
In some cases, as in the genus Bacillaria,.. ..this movement of sliding goes on till the frustules are on the point of separating, which then retrace their course till such a catastrophe seems equally imminent in the opposite direction.
We can see what Gosse means when we look at the video clip above and, for an explanation of these movements, he refers us to a paper by Wallich  who states:
The normal motion of the Diatomaceous frustule is in two opposite directions, which accord with its longest diameter. It is of a smooth, gliding nature, devoid of jerks or interruptions, and exhibits itself at tolerably regular intervals. The rate at which it travels is not uniform, being subject to variation on increase or diminution of light and warmth. The rate is also materially influenced by the condition of the endochrome, the motions being invariably more active and energetic when the frustule is full.
Wallich links the gliding movement to "elongated prehensile filaments", as particles are dragged along behind, or pushed in front of, the moving diatoms. We now know that motile diatoms, of which there are free-living as well as "colonial" types, move by exuding slime from what is termed a raphe. The slime produces a force against the substratum, or another diatom (should they be formed into a chain), that results in propulsion. At first sight, this might seem an expensive process, but the slime consists largely of carbohydrates that are generated in quantity during photosynthesis and these chemicals become much expanded when mixed with water. By exuding concentrated material through pores in the raphe, the rapid hydration then gives the observed propulsive force. The accurate observation by Wallich  that movement is associated with light and a "full" cell are evidence of an excess of carbohydrate from photosynthesis that can then be readily exuded.
When we observe gliding diatoms, we focus on the organisms, as we cannot see the slime that is produced, unless a chemical stain is used to show its presence. Interestingly, slimes from diatoms and many other organisms, from bacteria to mammals, have a most important role to play in the functioning of ecosystems .
For Henry Gosse, the beauty of diatom frustules, and the gliding movement shown by some of these algae, are further evidence of the wonder of God's Creation. For those of us that believe in evolution, there is a puzzle in thinking about how frustules developed through time and how the physiology of the algae evolved, allowing an excess of carbohydrates to be used as exudates that hydrate to produce a propulsive force. No-one would think that the variety of frustules evolved to please us and our approach to the puzzle is affected by our inability to understand the time over which evolution has occurred. Also, we don't know whether mutations resulted in a series of small changes or whether changes were on a larger scale. We can only speculate – and wonder.
 Philip Henry Gosse (1865) Land and Sea. London, James Nisbet & Co.
 G. C. Wallich (1860) Observations on the distribution and habits of the pelagic and freshwater free-floating Diatomaceae. The Annals and Magazine of Natural History 25: 1-20.
 Roger S. Wotton (2005) The essential role of exopolymers (EPS) in aquatic systems. Oceanography and Marine Biology: An Annual Review 42: 57-94.
The video of diatoms is by Donald Ott of the University of Akron and is available on YouTube at https://www.youtube.com/watch?v=NvHF-YjDZBo.