A recent post made me want to dig a bit deeper into the Euphorbia 'Diamond Frost' problem. I suppose more accurately, I wanted a place to house the many links I was generating regarding the history of how Euphorbia 'Diamond Frost' was named and maybe provide some clarity on what name should be used. Hopefully, this will act as a useful resource for anyone who also happens to be confused or annoyed by the problem.

Euphorbia 'Diamond Frost' has an odd history with taxonomy. It is commonly sold incorrectly as E. hypericifolia. 'Diamond Frost' is shown to be most closely related to E. graminea based on phylogentic and morphological evidence (E. graminea belongs to sect. Alectoroctonum while E. hypericifolia belongs to sect. Anisophyllum). One simple way to distinguish the two is to look at the lowest (non-bracteate) leaves. These will be opposite in E. hypericifolia and alternate in E. graminea (sometimes becoming whorled at the base of the inflorescence before becoming opposite when cyathia for the rest of the inflorescence; see pleiochasial and dichasial bracts). Problematically, the company that originally trademarked the name considers 'Diamond Frost' a hybrid, but what of is not specified. My best guess is that the company considers if a hybrid of cultivars rather than species.

Susan Mahr at the University of Wisconsin – Madison indicates that 'Diamond Frost' is a trademarked name that Euphorbia 'Inneuphdia' is commonly sold under. This cultivar was patented in 2005 under the name Euphorbia hypericifolia, though the trademarked name of 'Diamond Frost' is older. Given that it cites no prior cultivars and later cultivars cite Euphorbia 'Inneuphdia', I find it likely that this was the original patent that started the confusion. Many subsequent patents use Euphorbia or Chamaesyce hypericifolia but mention Euphorbia 'Inneuphdia'. One odd patent uses Euphorbia x hybrida (photo), but this is an intercultivar and not an interspecific hybrid as the Euphorbia 'Flame' used as the pollen parent is almost certainly Euphorbia 'Flameleaf' or another similar cultivar of E. graminea. Remarkably, one patent did provide the correct name.

Now, as for why 'Diamond Frost' is considered a "Euphorbia hybrid", I can't say for certain but know two things: 1. the initial patents provided names and 2. intercultivar crosses are close enough to hybrids in this confusing example. As such, I find it likely that the holders of the trademark recognized the taxonomic confusion and just went with Euphorbia hybrid to avoid controversy. It's just a hypothesis, but unless 'Diamond Frost' has changed since Euphorbia 'Inneuphdia' was initially patented, I find it the most likely outcome.

Next Tuesday (Feb 1st) from 12:00-1:00, I'll be giving a talk on Euphorbia sect. Anisophyllum. I'll talk about my Master's research and my experience integrating iNaturalist into sandmat taxonomy. If that sounds interesting to you, feel free to join!

Here's a link to the page: https://brit.org/events/euphorbia/

In case you missed it, here's a link to the YouTube video: https://www.youtube.com/watch?v=8Mp9NBkGP-w&feature=youtu.be

Here's the link: http://www.indefenseofplants.com/podcast/2020/3/15/ep-256-spurge-is-the-word
Hope you enjoy it!

Euphorbia abramsiana.

Many species of Euphorbia produce odd patterns on their leaves. Some of these have led to their production as ornamentals, while others are more random.

Maculations
Euphorbia maculations come in many shapes and sizes. They vary across species and across sections but seem to be primarily restricted to subgenus Chamaesyce. Within sect. Poinsettia and sect. Alectoroctorum, the maculations are simply spots scattered on the leaves without much pattern. They are most common in the dentata group including E. dentata, E. davidii, and E. cuphosperma among others. Euphorbia cuphosperma, in particular, produces maculations abundantly. A couple other species include E. heterophylla, and rarely E. graminea.

Euphorbia cuphosperma.

Euphorbia davidii.

Euphorbia heterophylla. Photo credit: Martin Reith

Euphorbia graminea. Photo credit: Ruth Ripley

Within sect. Anisophyllum, the maculations become more consistent both in presence and in pattern. The maculations become so consistent that they can actually be used for identification under some circumstances. The maculations tend to come in four basic patterns. It should be noted that maculations are inconsistent by their very nature, but the patterns they produce are often distinctive. Also, most species lack the ability to produce maculations. This makes maculations valuable as a characteristic if they are present, but essentially useless if they are absent. Also, most species within sect. Anisophyllum lose their maculations during the fall. This can be contrasted with the members of sect. Poinsettia which seem to generally keep them or even make them more pronounced in the fall.

Centralized and continuous
This is the most commonly encountered and the type found in E. maculata (spotted spurge). This type also occurs in species like E. glyptosperma, E. serpillifolia, E. serrula, E. velleriflora, E. hirta, E. ophthalimica (in Mexico), E. albomarginata, and some others.

Euphorbia maculata.

Subcontinuous
This is only known to occur in E. abramsiana and rarely E. hyssopifolia. The pattern is characterized by an irregular shape roughly along the midrib often with additional spots outside. It appears as though it is made up of many smaller spots. In E. abramsiana, pale lines in the position of pinnate venation are also generally present.

Euphorbia abramsiana.

Euphorbia abramsiana; maculation type verging on scattered.

Scattered
Found only in E. taluticola, some populations of E. pediculifera, and seedlings of E. hyssopifolia and E. nutans (in part; see proximally restricted). It is characterized by round spots scattered throughout the leaves like is occasionally found in sect. Poinsettia.

Euphorbia pediculifera. Photo credit: Mike Plagens.

Euphorbia hyssopifolia seedling displaying scattered maculation pattern. Photo credit: Zoologist123.

Proximally restricted
This occurs in E. hyssopifolia and E. nutans. I have not seen it in any other species, but it may occur in their relatives in Central and South America. This pattern is quite unique in that seedlings that produce maculation typically produce scattered patterns or sometime subcontinuous. Later midstem leaves have maculations that are positioned in the middle of the proximal two-thirds of the leaves. The maculation is generally continuous or subcontinuous, but is often wider than other species with these types of maculations. There is also often a white line along the midrib distal to the maculation.

Euphorbia nutans displaying broad, continuous maculation proximal to broad white line along midrib. Photo credit: Rick Travis.

Euphorbia hyssopifolia displaying broad, continuous maculation. Photo credit: Joshua Doby.

Euphorbia hyssopifolia displaying narrow, continuous maculations. Photo credit: odandno.

Euphorbia hyssopifolia seedling displaying scattered maculation pattern. Photo credit: Zoologist123.

Euphorbia hyssopifolia young plant displaying scattered/subcontinuous maculation pattern. Photo credit: James Bailey

Colored bracts
These are the most commonly noticed and frequently occur in sect. Poinsettia and in sect. Alectoroctonum. There are three basic forms this takes.

Gradation of color from base of leaf
This is commonly found in the dentata group but also occurs in E. heterophylla. This is perhaps most pronounced in E. mayfeildii and E. scheidiana. Euphorbia radians also probably best falls under this category even though the majority of bracts are usually fully light pink.

Euphorbia scheidiana. Photo credit: Leticia Soriano Flores.

Euphorbia radians. Photo credit: Oscar González.

Restricted color regions
Most pronounced and well known in E. cyathophora and E. marginata. Instead of a more gradual transition from green to white or some other color, the regions are well defined.

Euphorbia cyathophora; note the dark region between the green and red parts of the leaf. Photo credit: meldrake.

Euphorbia marginata.

Bracts fully colored
Most prominently seen in E. pulcherrima and related species. See also E. leucocephala and cultivars of E. graminea. If gradations occur, whole leaves are intermediate and less often producing intermediate colors on the same leaves.

Euphorbia pulcherrima.

Euphorbia leucocephala. Photo credit: Neptalí Ramírez Marcial.

Bands or other patterns
Some species produce patterns other than maculations on their leaves. These are rare and there are only a few species I have seen them in. Euphorbia graminea and members of the complex (sect. Alectoroctonum) sometimes produce bands similar to those of some Amaranthus species.

Euphorbia graminea. Photo credit: scottsuth.

Euphorbia xalapensis. Photo credit: solecito.

The other conditions are found in sect. Anisophyllum. Euphorbia abramsiana produces pale lines in the position of pinnate venation and E. hyssopifolia and E. nutans often produce white lines along the distal two-thirds of the midrib.

Euphorbia abramsiana.

Euphorbia nutans displaying broad, continuous maculation proximal to broad white line along midrib. Photo credit: Rick Travis.

Notes on anthocyanins
Maculations and red bracts are made of reddish pigments called anthocyanins. These are not just restricted to leaves, but may occur in every other structure. The key to understanding their significance is to know that there is typically a strong environmental influence in combination with the genetic influence. In maculations and bracts, the environmental influence is weaker. This makes them more useful as identification characteristics. There does appear to be some genetic variability in the anthocyanin production in other regions (see discussion on stems here and photo here). But with the exception of cyathia which really deserve their own post, the variability is generally unpredictable. They are mostly just gradations from green to red. Briefly, cyathia anthocyanin concentrations often vary based on the structure. The glands typically have the greatest concentration while the appendages have the least. The involucre and flowers (including ovaries and fruits) are typically inbetween, but this does vary.

The sandmat group (sect. Anisophyllum) of Euphorbia is quite different from the other groups. It is likely the most distinctive and has commonly been referred to by its own genus (most often and most recently Chamaesyce). It is probably most commonly known in North America for spotted spurge. I have written fairly extensively and technically about the differences, but I'm going to attempt to simplify it all here. If you are interested in the more technical explanation, you might try reading the morphology section of my thesis.

Spotted spurge (Euphorbia maculata).

C4 photosynthesis and Kranz anatomy
Perhaps the most consistant and interesting charactaristic is the presence of C4 photosynthesis, which occurs in all but subsect. Acutae (i.e., Euphorbia acuta, E. angusta, and E. johnstonii). Of course, you can't really see C4 photosynthesis, but there is a cool trick you can use to tell if a plant has C4 or C3 photosynthesis. C4 photosynthesis comes with specialized anatomy called Kranz anatomy where the primary photosynthesizing cells are clustered around the veins. This makes the veins look darker than the surrounding tissue creating an interesting pattern.

Kranz anatomy in Euphorbia prostrata (left). Euphorbia angusta, the only species in section Anisophyllum confirmed to have C3 photosynthesis (right).

Dorsi-ventral (sided) stems and unequal leaves
In most species, there is a clear upper and lower side to the stems (i.e., a sidedness; also known as dorsi-ventral because the stems have a dorsal and ventral side). This is most obvious in prostrate species but is present in most upright species too. Take for instance hyssop spurge (E. hyssopifolia), nodding spurge (E. nutans), or graceful spurge (E. hypericifolia). In all these cases, the stems are generally quite upright, but the stems bend toward the tips so that the upper sides of the branches face the sun (usually described as arcuate ascending). This sided appearance comes in part from the asymmetric nature of the leaves. One side of the leaf (as divided by the midvein) is almost always longer than the other side. This makes the leaves more efficient when they turn sideways and allows the lobe of the longer side (what I have sometimes referred to as the cordate lobe) to overlap the stem, but the other side not to be obscured by the stem. The plants are so well evolved for a prostrate lifestyle that their leaves are built to make the most of it. There are rare exceptions to this rule (look at E. acuta for example), but it is a quick and pretty reliable technique to distinguish from the other groups.

Arcuate ascending habit and dorsi-ventral positioning in E. hypericifolia (left). Greatly exagerated assymetric leaves in a E. glyptosperma seedling (right).

Non-glandular stipules
Stipules are structures made of leaf tissue that grow on either side of a petiole. In some species outside Euphorbia, these can become leafy, thorny, or even twining (some tendrils are modified stipules). In Euphorbia they are almost always small, inconspicuous structures or may be absent altogether. In sect. Anisophyllum, they are generally small and white and can be found between the pairs of leaves. Each leaf produces one on each side of the stem (one on the dorsal, one on the ventral). Each individual stipule may become lobed to the point of appearing like more than one structure or can fuse with the other leaf's stipule on its side of the stem to form a scale. Though this can be useful for ID within sect. Anisophyllum, the key is that they are almost always present and almost never glandular. This is what helps to distinguish them from other groups.

Laciniate stipules of Euphorbia glyptosperma (left). Stipules of E. albomarginata that are fused together into a single white scale (right).

Sympodial branching (pseudo-dichotomous)
The branching is also interesting in sect. Anisophyllum. Within the group, each branch produced terminates (either stops growing, aborts, or is terminated by a cyathium) and is followed up by a pair of branches. This looks very similar to dichotomous branching which produces the same forking at the nodes, but this is slightly different. In true dichotomous branching (which can be seen in the wisk fern, genus Psilotum), the apical bud itself splits into two equal buds. In sympodial branching, the apical meristem itself stops and branches take over in its sted. This has led to the term pseudo-dichotomous branching (i.e., false dichotomous branching).
For distinguishing from other members of Euphorbia, the key is that members of sect. Anisophyllum have this branching right down to the base. Its very first apical meristem produces one pair of leaves and produces nothing else. The branches take over from there. In other Euphorbia groups, the inflorescences are often sympodial, but the vegitative stems are not. This has led to speculation that members of sect. Anisophyllum are the result of reduction of plants to only the inflorescence but this explaination as it has been traditionally proposed has faced some critisism, so I will not go into the details here. If anyone wants to learn more, I recommend Hayden (1988): Ontogeny of the Cotyledonary Region of Chamaesyce maculata (Euphorbiaceae). Regardless, this is another useful characteristic!

Sympodial branching and equal length in paired branches of E. golondrina (left). Seedling of E. glyptosperma showing blank, flat spot where the apical meristem would be in most other plants (right).

Ecarunculate seeds
Lastly, members of sect. Anisophyllum have seeds that aren't carunculate. A caruncle is an outgrowth at the part of the seed that attaches to the fruit called the hilum. Many members of Euphorbia have carunculate seeds, but not members of sect. Anisophyllum. The closest they come is in the species E. carunculata but despite the name, the enlarged structure is an outgrowth of the micropyle (the part of the seed where the pollen grew through), not the hilum. For this one, it's probably easier to explain with pictures than words. If you want the full explanation or to know how seeds can be used in identification, I recommend reading my advanced seed morphology explanation.

Ecarunculate seed of E. lata ("E" represents the hilum), sect. Anisophyllum (left). Carunculate seed of E. tetrapora, subgenus Esula (right).

Outgrowth of the micropyle in E. carunculata, sect. Anisophyllum (left).

Taxonomically significant characteristics
These are characteristics you're likely to encounter when trying to key out members of this section.

Habit
There are several major forms. In the US, only the Hawaiian endemics, E. jaegeri, and E. chaetocalyx var. triligulata are woody (and they very from shrubs to trees). Euphorbia jaegeri, and E. chaetocalyx var. triligulata are subshruby or barely so while the Hawaiian endemics are much larger (some becoming trees). With herbaceous species, some have woody rootstocks, while some have non-woody rootstocks. This characteristic is significant outside the southernmost parts of the southern US deserts for differentiating perennials from annuals, but less so nearer the southern boarder. Among annuals, there are two main forms with intermediate species filling in the gaps between them. One is exemplified by the prostrate form like E. maculata or E. prostrata. The other is robust and upright and best exemplified in E. hypericifolia, E. hyssopifolia, and E. nutans.

Much branched growth from woody rootstock of E. fendleri (left). Typical annual growth of E. golondrina (center). Robust upright growth of E. hyssopifolia (right).

Inflorescences
Cyathia (flower-like inflorescences; more information here) may be densely clustered or solitary at the stems. Also, they may be congested along the stems or compacted into a round structure (i.e., a glomerule). There are gradations between all these characteristics but most keys try to distinguish between glomerules and other forms typically termed solitary.

Solitary cyathia of E. acuta (left). Conjested cyathia in elongated, leafy inflorescences of E. stictospora (center). Glomerules of E. hypericifolia (right).

One useful comparison to make when determining whether cyathia are solitary or in glomerules is to compare E. hyssopifolia and E. hypericifolia along with typical solitary cyathia. The glomerules are ultimately formed by shortining of the internodes after cyathial production and an eventual elimation of leaf production. An extra step of stems branching in the dorsal or ventral direction to the main axis to form a three-dimensional glomerule is sometimes taken in larger inflorescences. This can be contrasted with the two-dimensional plane in which all cyathia occur when cyathia are solitary.

Solitary cyathia of E. acuta (left). Inflorescences of E. hyssopifolia (center). Glomerules of E. hypericifolia (right).

The differences between the two congested inflorescence types are simply a matter of branching. In sect. Anisophyllum, branching is sympodial (pseudo-dichotomous) as mentioned above. This means that each branch forked into two (though one may be dominant over the other). In the glomerules, the forked branches are of essentially equal lengths. In the other dense inflorescence type, it appears that one branch is dominant over the other leading to what looks like a central axis with a proliferation of cyathia. In the case of inflorescences like E. hyssopifolia, the main distinction is in whether the inflorescence has leaves or not, or at least if it appears to. What the research is typically referring to is whether there are several leaves for the number of cyathia or few to none. As with many characteristics, it's rarely an absolute thing, but makes sense and is quite characteristic if you look at several examples.

Stem and fruit hairs
Depending on group, stem and fruit hairs may be completely unreliable as taxonomic characteristics or among the most reliable. In the US, the former case is rare and the latter is generally true. Examples where the presence or absence of hairs are unreliable include: Euphorbia polycarpa (all varieties), E. micromera, and California populations of E. serpillifolia. These plants are mostly found in the Sonoran Desert, Mojave Desert, and adjacent areas (the absence of hairs of E. micromera in Texas and New Mexico is actually a stable characteristic). Some examples where hairs may be present or absent outside of this area can be found in E. hyssopifolia, E. villifera, and sometimes E. abramsiana (also, potentially E. serrula in Mexican populations). Presence or absence of hairs is most often used, but sometimes the type of hairs is also important. For instance, when hairs are present in E. hyssopifolia, they are pilose (straight and longer) while the hairs of E. nutans are very short and crisped. Another common example: E. maculata has appressed fruit hairs while those of E. prostrata are pilose (except in rare cases where excessive sun exposure and/or trampling are involved).

Dust-like curly hairs of E. nutans (left). Hair stems and glabrous fruits of E. serrula (center). Hairy stems, fruits, and leaves of E. maculata (right).

Section in progress

Glands and appendages

Fruit shape

Styles

E. deltoidea var. adhaerens

Photo credit (left): Jay Keller (click here for observation). Photo credit (right): Jay Keller (click here for observation).

iNaturalist observations, Flora of North America.

E. deltoidea var. deltoidea

Photo credit (left): Pablo L Ruiz (click here for observation). Photo credit (right): Jay Keller (click here for observation).

iNaturalist observations, Atlas of Florida plants, Flora of North America.

E. deltoidea var. pinetorum

Photo credit (left): Jay Keller (click here for observation). Photo credit (right): Jay Keller (click here for observation).

iNaturalist observations, Atlas of Florida plants, Flora of North America.

E. deltoidea var. serpyllum

Photo credit (left): Jay Keller (click here for observation). Photo credit (right): Jay Keller (click here for observation).

iNaturalist observations, Atlas of Florida plants, Flora of North America.

Flora of North America treatment to species.

These plants have serrated leaves and glands without appendages. According to Euphorbia PBI, there are about 180 taxa. To go through them all would be very time consuming and well beyond the scope of this project. However, in order to better understand the species that have been introduced to other locations, it is worth discussing a few outside the US that have spread outside of Afroeurasia where most species are native. Notes on leaves refer to the lower leaves and usually includes the similar pleiochasial bracts, but not the dichasial bracts (click here for notes on the bracts). Plants known to occur in New Zealand are marked "NZ" at the end. Plants known to occur in the United States are marked "US" at the end.

E. oblongata - Perennial; stems villous; leaves glabrous, ranging from oblong, elliptic, to broadly lanceolate or ovate, not linear or linear-lanceolate, typically broadest at the base or middle but may be broadest at the apices, apices rounded; glands typically 2-3 (typically 2 in dichasia); fruits warty. Originally from Eurasia, but introduced elsewhere. NZ US
E. depauperata - Perennial; stems glabrous or pilose; leaves glabrous or occasionally pilose, typically linear-lanceolate, sometimes slightly broader, apices typically acute; glands 4-6; fruits warty. Africa.
E. epithymoides - Perennial; stems villous; leaves pilose; broadest towards the middle or base, apices rounded; glands 2-5 (even in same inflorescence) but most commonly 4; fruits covered with many narrow, hair-like warts. Originally from Eurasia, but introduced elsewhere. US
Annuals - Many of these are quite difficult to distinguish and may require seed characteristics.
E. platyphyllos - Annual; stems usually glabrous; leaves glabrous or hairy, usually oblanceolate, typically broadest at apices, apices acute; many dichasia produced before terminal pleiochasium of 3-5 branches; glands 4; fruits warty; seeds smooth, over 2 mm long. Two forms: 1. plants glabrous, more common. 2. plants hairy, less common (specimen 1; photos). Type much more robust than most specimens representing the species. Originally from Eurasia, but introduced elsewhere. NZ US
E. stricta - Annual, glabrous; leaves usually oblanceolate, typically broadest at apices, apices acute; many dichasia produced before terminal pleiochasium of 3-5 branches; glands 4; fruits warty; seeds smooth, under 2 mm long. Two forms: 1. leaves sharply acute, margins sharply serrulate; dichasial bracts mucronate; fruits warts shorter (specimen 1; specimen 2). 2. leaves less sharply acute, margins less sharply serrulate; dichasial bracts mucronate or commonly not; fruit warts long (photos; specimen). Misidentifications? Originally from Eurasia, but introduced elsewhere. NZ
E. spathulata - Annual, glabrous; leaves usually oblanceolate, typically broadest at apices, apices acute; glands 4; fruits warty, warts short; seeds reticulate. Seemingly restricted to North America, widespread. US
E. alta - Annual, glabrous; leaves usually oblanceolate, typically broadest at apices, apices acute; glands 4; fruits warty, warts long; seeds reticulate; restricted to SW United States and NW Mexico. US
E. texana - Annual, glabrous; leaves usually oblanceolate, typically broadest at apices, apices acute; glands 4; fruits smooth; seeds reticulate. Restricted to E Texas and Louisiana in the United States. US
E. helioscopia - Annual, glabrous; leaves usually obovate, broadest at apices, apices rounded or truncate; glands 4; fruits smooth. Originally from Eurasia, but introduced elsewhere. NZ US

Other species:
E. apios: link 1
E. hirsuta: link 1
E. palustris:
E. pterococca: link 1
E. valerianifolia: Fruits sparsely long pilose; link 1

Notes on E. oblongata/depauperata.

Section Alectoroctonum is a section of New World Euphorbias that is generally distinguished from many other groups by having petal-like appendages and not possessing any particular specialized characteristics. The group is quite diverse and may form herbs, shrubs, or even succulents (see E. antisyphilitica). They are so diverse that the seem to hold the middle-ground between many of the better-defined groups of US Euphorbias like sect. Anisophyllum and subg. Esula. They generally have dichasial branching like members of subg. Esula, but the bracts are generally similar enough to the stem leaves that they evade notice (except in species like E. marginata). They generally have alternate leaves below, but some have lower stems so reduced so as to only produce opposite leaves (see E. macropus). Furthermore, the group may be confused with members of sect. Nummulariopsis. Section Alectoroctonum is best learned by simply learning the main species in the group and comparing to others outside the group. There is some continuity, but it is difficult to define.

Here, I am considering the species in states east of Texas. Observations from the area. Click on the species name in bold to see observations from the area verified by me.

Euphorbia marginata

Photo credit: Sam Kieschnick (click here for observation).

Euphorbia graminea

Photo credit (left): Jay Keller (click here for observation). Photo credit (right): Jay Keller (click here for observation).

Euphorbia polyphylla

Photo credit (left): Jay Keller (click here for observation). Photo credit (right): Jay Keller (click here for observation).

Euphorbia ipecacuanhae


Photo credit (upper left): Rob Van Epps (click here for observation). Photo credit (upper right): Michael Ellis (click here for observation).

Photo credit (lower left): Jason Hafstad (click here for observation). Photo credit (lower right): Joshua Tewksbury (click here for observation).

Euphorbia exserta

Photo credit (left): ericpo1 (click here for observation). Photo credit (right): Jay Keller (click here for observation).

Euphorbia mercurialina

Photo credit (left): Erin Faulkner (click here for observation). Photo credit (right): Jonathan (JC) Carpenter (click here for observation).

Euphorbia curtisii


Photo credit (upper left): Andy Newman (click here for observation). Photo credit (upper right): jtuttle (click here for observation).

Photo credit (lower left): whiteoak (click here for observation). Photo credit (lower right): whiteoak (click here for observation).

Euphorbia hexagona

Photo credit: Brush F (click here for observation).

Euphorbia corollata complex
The photos bellow appear to represent true examples of their respective species. However, many populations are very difficult to determine. Some seem intermediate while others seem to represent distinct entities. There is at least one undescribed species (photos here) represented in this group.

Euphorbia discoidalis

Photo credit (upper left): cwarneke (click here for observation). Photo credit (upper right): howardhorne (click here for observation).

Euphorbia pubentissima

Photo credit: Janet Wright (click here for observation).

Euphorbia corollata


Photo credit (upper left): Janet Wright (click here for observation). Photo credit (upper right): Alvin Diamond (click here for observation).

Photo credit (lower left): Kit Howard (click here for observation). Photo credit (lower right): Nathan Taylor (click here for observation).

References to look into:
Monograph of Euphorbia sect. Tithymalopsis(Euphorbiaceae)
A New Species of Euphorbia Subgenus Chamaesyce Section Alectoroctonum (Euphorbiaceae) From Limestone Hills of Wayne County, Mississippi
Seed morphology ofEuphorbia sectionTithymalopsis (Euphorbiaceae) and related species
Pollen morphology of Euphorbia subgenus Agaloma section Tithymalopsis and related species (Euphorbiaceae)
Isozyme and Morphological Divergence Within Euphorbia Section Tithymalopsis (Euphorbiaceae)

Seed morphology is very important to the taxonomy of the genus Euphorbia. It is so important that most larger keys require looking at the seeds. The seeds are so diagnostic that there are many instances where species can be determined just by looking at the seeds alone. Because of this, I think it's time that I write a little bit about the more complicated aspects of Euphorbia seed morphology. This does not cover the full variability of seed morphology but is intended to give an introduction to help you understand the terminology.

In almost every Euphorbia seed, there are four more or less flat sides (referred to here as faces). Two of these are generally longer than the other two as shown below.

These are useful for identification, but in order to understand the morphology of the seeds, there are some other characteristics that should be taken into account. First off, the faces are best identified in their relationship to the four angles. Look at the figures below.

Left and Right: A: distal angle (angle furthest from center of fruit); B: lateral angles; C: proximal angle (angle nearest center of fruit; also called the raphe). Left only: D: testa (seed coat); E: mucilaginous coat (only the white part). Right only: D: chalazal end; E: hilum (micropyle visible as brown dot directly right of hilum).

The two faces adjacent to the distal angle are known as distal faces. The two angles adjacent to the proximal angle are known as proximal faces. There is considerable variation in these faces between species. In some, the faces bulge so much or the angles compress so much that the faces are nearly indistinguishable from the angles and the seeds become essentially round in cross-section. In some, these faces are compressed and create somewhat deep depressions between the angles. The topography of the angles varies too. They may be ridged, wrinkled, pitted, papillate, or have some variation of these. In some species, it is important to know whether the ridges overlap the angles. Some have sharper ridges than others.

Aside from faces and angles, the figures above also identify the testa (the seed coat), the mucilaginous coat (the part that makes the coat slimy/sticky when wet), the chalazal end (defined as the end of the raphe opposite to the hilum), the hilum (place where the seed was attached to the fruit), and the micropyle (place where the pollen entered into the ovule during fertilization). Many of these exhibit variability between species that make them diagnostic. A characteristic not shown in the above figure is a caruncle (an outgrowth of the hilum), which can be useful for identification.

Examples:

Euphorbia carunculata with an outgrowth of the micropyle. Source observation.

Euphorbia lata showing connections between hilum, micropyle, and fruit tissue. Source observation.

Euphorbia alta seed with caruncle. Source observation.

Euphorbia roemeriana displaying caruncle and enlarged chalaza. Source observation.

Seeds of various species showing diversity in ridges. Top left: E. serrula; top right: E. simulans; bottom left: E. prostrata; bottom center: E. glyptosperma; bottom right: Euphorbia theriaca var. theriaca.

I've heard it mentioned a couple of times that it helps a lot for people to have an idea of what they need to photograph. Hopefully, this will help.

1. Habit.
This is tied for the most important thing to photograph. If the photo is clear, all the essential characteristics will be visible in one or two good habit shots. In plants that lie flat along the ground, this only needs to be a top-down photo. In something that is upright, it's important to get a top-down photo and one from the side or potentially one at an angle in-between.

2. Branches.
Stems with leaves, flowers, and fruit if present. This one is just as important as habit. This allows for seeing a lot of major characteristics including leaves, phyllotaxy, stipules, cyathia, and sometimes fruit.

3. Undersides of leaves and/or branches (generally optional).

This helps with some groups and may help one see styles which are important in a few species. This can be particularly helpful for the species that lie flat along the ground.

Close-ups of the characteristics themselves (all usually optional):

Stems.
For some species, you may need information about the hairs on the stems or even the stems themselves. To facilitate this, a close-up of the stem can be helpful. For instance, one useful way to distinguish E. hyssopifolia from E. nutans is be looking for internode hairs (long and straight or absent in the former; short and curly in the latter). Stems can also be winged (E. serpillifolia), somewhat flattened or completely round.
Species where this level of detail may be helpful: E. hyssopifolia, E. nutans, E. dentata, E. davidii, E. serpillifolia, and E. glyptosperma.

Leaves.
Depending on what group you photograph, there may be more than one type of leaf. This is particularly important in subgenus Esula, which have the typical stem leaves and then two types of bracts. Many members outside subgenus Esula have leaves that are different when growing under a cyathium. When this is the case, it can be useful to photograph all different types. It's always best to get photos of both sides of the leaves, though usually not necessary in the US members of the genus Euphorbia.
Species where this level of detail may be helpful: members of subgenus Esula, E. bicolor, E. marginata, possibly E. dentata, and possibly E. davidii.

Stipules.
Small triangular (often membraneous) structures between two pairs of leaves in section Anisophyllum. Stipules are technically made from leaf tissue at the base of the petiole. Stipules vary from being entire to deeply lobed or even fused with its neighboring stipule. These may be hairy or non-hairy. In some, they are even reduced to simple glands.
Species where this level of detail may be helpful: members of section Anisophyllum.

Cyathia.
Morphology explained simply here. Diversity in cyathial form within Euphorbia is vast. Though not always necessary for an ID, the uniqueness of these structures make them fun to photograph. Nearly every aspect of the cyathium may differ between species, but the most readily visible and often most useful diversity can be found in the glands and their appendages. The number of glands can vary from 1 to 5, and typically not 2 or 3 (rarely, plants will produce 2, 3, 6 and very rarely more glands). The gland ornaments can be petal-like, horn-like, or absent. If petal-like, they can be divided or undivided. They are usually white but can be pink, red, or (if horn-like) even yellow. These characteristics are usually visible from the branch photos but are fun to look at more closely.
Species where this level of detail may be helpful: essentially all, though especially E. arizonica, E. setiloba, E. spathulata, E. texana, E. ocellata and look-alikes, E. missurica, E. parryi, and eastern species of section Alectoroctonum.

Fruits and styles.
The fruits are not quite as variable as cyathia but are no less important. Fruits may be hairy or not hairy; warty, papillose, or smooth; and/or deeply lobed or subtriangular in cross-section. The three styles found at the apex of the fruit and maturing ovaries usually have two branches which may be divided all the way to the base, not divided at all, or be divided somewhere in-between.
Species where this level of detail may be helpful: essentially all.

Seeds.
Though difficult to photograph and only completely essential to identification in a couple of species pairs in the US, the seeds can be really important to identification. If you are unfamiliar with the group and have to key them out, you essentially must look at seeds. Seeds may be smooth, ridged, papillose, pitted, carunculate, ecarunculate, quadrangular, or round. Tips for extracting and photographing seeds can be found here.
Species where this level of detail may be helpful: essentially all, though especially E. simulans, E. theriaca, E. hyssopifolia, E. nutans, E. serpillifolia, E. glyptosperma, E. dentata (preferably with scale), E. davidii (preferably with scale), E. pubentissima (with scale), and E. corollata (with scale).